DE102017111926A1 - Process control circuit and method for controlling a processing arrangement - Google Patents

Abstract

According to various embodiments, a processing circuit (108) may include: a data acquisition circuit (102) configured to acquire process data and measurement data of a processing device (104), wherein the process data includes a plurality of process event data according to a plurality of process events and the measurement data includes a plurality of measurement event data corresponding to a plurality of measurement events contain; a data merge circuit (108) configured to provide state information, wherein the state information represents a status of the processing device (104), and provide accuracy information representing an accuracy of the state information; wherein the data fusion circuit (108) is further configured to update the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and to restore the prior state of state information independently of updating the state information based on the acquired measurement event data Status information, and a control circuit (110) configured to control the processing device (104) based on the state information and / or based on the accuracy information.

Description

Technical area

Various embodiments generally relate to a process control circuit and method for controlling a processing arrangement.

background

In general, more or less complex processing arrangements can be used in many industries to produce the desired goods, e.g. As electronic devices, mechanical devices, electromechanical devices and the like, manufacture or edit. Such processing arrangements may include multiple processing tools or any other suitable apparatus for processing the goods. In the semiconductor industry, a so-called tool park containing multiple machining tools may be used to machine any suitable type of workpiece, e.g. For processing a wafer, a chip, a chip package, a circuit board, and the like. In the semiconductor industry, machining a workpiece may include one or more layering processes, one or more patterning processes, one or more temperature treatments, one or more implantation processes, and so on. Although the editing tools may be individually controlled, a global control system may be used to control all editing tools, e.g. B. for setting the respective processing tools, for setting one or more operating parameters of the respective processing tools or for a different setting of the respective processing tools. The local and / or global (eg, factory-wide) control of the respective machining tools may also be referred to as a run-to-run system. In general, however, it may be difficult to control a large number of editing tools that also perform a large number of different data processing operations.

Summary

According to various embodiments, a processing circuit may include: a data acquisition circuit configured to acquire process data and measurement data of a processing device, the process data including a plurality of process event data according to a plurality of process events and the measurement data including a plurality of measurement event data according to a plurality of measurement events; a data merging circuit configured to provide state information, wherein the state information represents a status of the processing device, and to provide accuracy information representing an accuracy of the state information; wherein the data fusion circuit is further configured to update the state information based on the acquired process event data, thereby decreasing the accuracy of the state information, and independently of updating the state information, to restore the previous state of the state information based on the acquired measurement data ("downdate"), whereby the accuracy of the state information is increased, and a control circuit configured to control the processing arrangement based on the state information.

list of figures

• 1 eine schematische graphische Darstellung einer Prozesssteuerschaltung gemäß verschiedenen Ausführungsformen zeigt,
• 2A und 2B ein Beispiel einer Operation einer Datenverschmelzungsschaltung gemäß verschiedenen Ausführungsformen zeigen,
• 3A eine schematische graphische Darstellung des Betriebs einer Prozesssteuerschaltung gemäß verschiedenen Ausführungsformen zeigt,
• 3B eine schematische graphische Darstellung des Betriebs einer Prozesssteuerschaltung gemäß verschiedenen Ausführungsformen zeigt,
• 4A bis 4E verschiedene Konfigurationen einer Datenverschmelzungsschaltung in einer schematischen graphischen Darstellung gemäß verschiedenen Ausführungsformen zeigen,
• 5A und 5B ein Beispiel einer Modellmehrdeutigkeit in einer schematischen graphischen Darstellung gemäß verschiedenen Ausführungsformen zeigen,
• 6A bis 6I verschiedene Konfigurationen einer Datenverschmelzungsschaltung gemäß verschiedenen Ausführungsformen zeigen,
• 7A und 7B verschiedene Konfigurationen einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigen,
• 8 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 9 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 10 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 11 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 12 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 13 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 14 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 15 eine Konfiguration einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigt,
• 16A bis 16C verschiedene Konfigurationen einer Datenverschmelzungsschaltung gemäß verschiedenen Ausführungsformen zeigen,
• 17A und 17B verschiedene Konfigurationen einer Datenverschmelzungsschaltung in einer schematischen Ansicht gemäß verschiedenen Ausführungsformen zeigen, und
• 18 einen schematischen Ablaufplan eines Verfahrens zum Betreiben einer Prozesssteuerschaltung gemäß verschiedenen Ausführungsformen zeigt.

In the drawings, like reference characters generally refer to the same parts throughout the several views. The drawings are not necessarily to scale, with emphasis instead being generally placed upon illustrating the principles of the invention. In the following description, various embodiments of the invention will be described with reference to the following drawings, wherein:

• 1 FIG. 2 shows a schematic diagram of a process control circuit according to various embodiments, FIG.
• 2A and 2 B show an example of an operation of a data fusion circuit according to various embodiments,
• 3A FIG. 2 shows a schematic diagram of the operation of a process control circuit according to various embodiments, FIG.
• 3B FIG. 2 shows a schematic diagram of the operation of a process control circuit according to various embodiments, FIG.
• 4A to 4E show various configurations of a data fusion circuit in a schematic diagram according to various embodiments,
• 5A and 5B show an example of a model ambiguity in a schematic diagram according to various embodiments,
• 6A to 6I show various configurations of a data fusion circuit according to various embodiments,
• 7A and 7B show various configurations of a data fusion circuit in a schematic view according to various embodiments,
• 8th shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 9 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 10 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 11 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 12 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 13 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 14 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 15 shows a configuration of a data merging circuit in a schematic view according to various embodiments,
• 16A to 16C show various configurations of a data fusion circuit according to various embodiments,
• 17A and 17B show various configurations of a data fusion circuit in a schematic view according to various embodiments, and
• 18 12 shows a schematic flowchart of a method for operating a process control circuit according to various embodiments.

description

The following detailed description refers to the accompanying drawings, which, for purposes of illustration, show specific details and embodiments in which the invention may be practiced. The embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be used, structural, logical, and electrical changes being made without departing from the scope of the invention. The various embodiments are not necessarily mutually exclusive, as some embodiments may be combined with one or more other embodiments to form new embodiments. Various embodiments are described in the context of methods, while various embodiments are described in the context of devices. However, it will be appreciated that the embodiments described in connection with methods may apply similarly to the devices, and vice versa.

The terms "at least one" and "one or more" may be understood to include any integer greater than or equal to one, d. h., one, two, three, four, [...], etc., included. The term "multiple" may be understood to mean any integer greater than or equal to two, that is, any number greater than or equal to two. h., two, three, four, five, [...], etc., contains.

The phrase "at least one of" with respect to a group of elements may be used herein to mean at least one element of the group consisting of the elements. The phrase "at least one of" with respect to a group of elements can be used here for. B. may be used to mean a selection of: one of the listed items, a plurality of one of the listed items, a plurality of individual listed items, or a plurality of multiple listed items.

As used herein, a "circuit" is understood to be any type of logic-implementing entity that may include specialized hardware or a processor that executes software. A circuit may thus comprise an analog circuit, a digital circuit, a composite signal circuit, a logic circuit, a processor, a microprocessor, a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), a field programmable gate array (FPGA), an integrated circuit, an application specific integrated circuit (ASIC), etc., or any combination thereof. Any other way of implementing the respective functions, which will be described in more detail below, may also be understood as a "circuit". It will be appreciated that two (or more) of the circuits described in detail herein may be implemented as a single circuit having substantially equivalent functionality, and conversely that any single circuit described in detail herein may be implemented as two (or more) separate circuits having a substantially equivalent functionality can be realized. In addition, the references to a "circuit" may refer to two or more circuits that together form a single circuit. The term "circuitry" may refer to one or more circuits and / or an electronic device consisting of one or more circuits.

As used herein, a "processing circuit" (or equivalent, "processing circuitry" or "processor") is understood to refer to any circuit that performs an operation (s) on a signal (s), such as a device. For example, any circuit that performs processing on an electrical signal or an optical signal. A processing circuit may thus refer to any analog or digital circuitry that changes a characteristic or property of an electrical or optical signal that may include analog and / or digital data. A processing circuit may thus be referred to an analog circuit (which may be explicitly referred to as an "analog processing circuit"), a digital circuit (which may be explicitly referred to as a "digital processing circuit"), a logic circuit, a Processor, a microprocessor, a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), a field programmable gate array (FPGA), an integrated circuit, an application specific integrated circuit (ASIC), etc., or any combination thereof. Accordingly, a processing circuit may refer to a circuit that performs processing on an electrical or optical signal as hardware or software such. B. executes software executed in hardware (eg, a processor or a microprocessor). As used herein, "digital signal processing circuitry" may refer to a circuit implemented using digital logic that is capable of processing on a signal, e.g. An electrical or optical signal, comprising a logic circuit (s), a processor (s), a scalar processor (s), a vector processor (s), a microprocessor (s), a controller, a microcontroller, a Central processing unit (s) (CPU), graphics processing unit (s) (GPU), digital signal processor (s) (DSP), field programmable gate array (s) (FPGA), integrated circuit (s), application specific integrated circuit (s) ) (ASIC) or any combination thereof. Furthermore, it is recognized that a single processing circuit may equivalently be divided into two separate processing circuits, and conversely, two separate processing circuits may be combined in a single equivalent processing circuit.

The following provides a general description of a process control circuit according to various embodiments, followed by various embodiments describing the process control circuit in greater detail. 1 illustrates a process control circuit 100 in a schematic diagram according to various embodiments.

The process control circuit 100 can be a data acquisition circuit 102 contain. The data acquisition circuit 102 can be configured, the process data 105 and the measurement data 107 a processing arrangement 104 capture. The processing arrangement 104 can have multiple editing tools 105t , z. As manufacturing tools, or contain any other type of processing structures. Furthermore, the processing arrangement 104 several measuring tools 107T or any other type of measurement structures.

According to various embodiments, the process data 105 multiple process event data 105d according to several process events 105e contain. The measured data 107 can have multiple measurement event data 107d according to several measurement events 107e included (see 3 ).

According to various embodiments, the processing tools 105t contain any tool used in semiconductor processing, e.g. A lamination tool (eg, performing chemical or physical vapor deposition), a pattern generation tool (eg, a lithography tool), an etch tool, a grinding tool, a die cut tool, a glow tool, and the like. According to various embodiments, the measuring tools 107T contain any tool used in semiconductor processing, e.g. A tool for measuring a thickness of a carrier, a tool for measuring optical characteristics, a tool for measuring electrical characteristics, a tool for testing a machined structure, and the like.

In the case of semiconductor processing, any of the processing tools used 105t the processing arrangement 104 the process event data 105d for a process event 105e Deploy, where is a process event 105e z. B. may be any type of process associated with the workpiece being processed, e.g. B. on a wafer, a chip, an assembly and the like, related machining tool 105t is performed. Furthermore, any of the measuring tools used 107T the processing arrangement 104 the measurement event data 107d for a measurement event 107e provide a measurement event 107e z. For example, any type of one with the on the machined workpiece, for. B. on a wafer, a chip, an assembly and the like, related measuring tool 107T can be performed measurement. Therefore, according to various embodiments, for a respective workpiece, W that with the editing arrangement 104 a corresponding set of correlated process data and measurement data is provided, W ₁ ( 105 - 1 . 107 - 1 ) W ₂ ( 105 - 2 . 107 - 2 ). The correlated process data and measurement data can be used to control the processing arrangement 104 in any desired manner, the processing arrangement 104 a large number of editing tools 105t and / or measuring tools 107T , z. B. more than 100 Machining tools and / or measuring tools, which may also be referred to as pass-to-pass control. Due to the combination of the large number of tools and the large number of different workpieces (or illustrative products), a huge amount of data has to be processed.

Furthermore, the process control circuit 100 a data fusion circuit 108 that is configured, state information, X to provide. The state information represents a status of the processing device 104 , where the accuracy information .DELTA.X an accuracy of the state information X represent. According to various embodiments, the generation of the accuracy information .DELTA.X be important to the editing arrangement 104 based on the collected data 105 . 107 to steer properly. The processing data 105 may also include model data or model parameters for a corresponding process model for modeling a process of one or more of the multiple processing tools 105t and / or measuring tools 107T contain. According to various embodiments, the processing arrangement 104 a database circuit or any other suitable circuit for providing the process data 105 and / or the measured data 107 contain.

According to various embodiments, the data fusion circuit 108 Further, the state information may be configured X (eg, by performing a process update) based on the captured process event data 105 to update, thereby improving the accuracy of the state information X is decreased, and regardless of updating the state information, the previous state of the state information X based on the collected measurement event data ( 107 ) (eg, by performing a restore of the previous state, also referred to as a metrology update), thereby increasing the accuracy of the state information X is enlarged. Illustratively, the state information X including as much information as possible (ie, based on the process data 105 and the measurement data 107 ), which are available at that time, are kept current. The state information X may also allow a result of possible subsequent process events 105e and / or measurement events 107e predict. Therefore, the processing arrangement 104 based on the state information X to be controlled. Alternatively or additionally, the processing arrangement 104 based on the accuracy information .DELTA.X to be controlled.

Furthermore, the process control circuit 100 a control circuit 110 that is configured, the editing arrangement 104 based on the state information X and / or based on the accuracy information .DELTA.X to control 111 ,

According to various embodiments, the data fusion circuit 106 be configured, the processes of the process arrangement 104 including the accuracy data for all of the process data 105 and the measured data 107 to model.

According to various embodiments, the process control circuit 100 according to various embodiments, as described below. The embodiments may be used alone or in combination in the process control circuit 100 be included. Various embodiments relate to the proper use of the accuracy information .DELTA.X to the editing arrangement 104 to control and / or any other connected circuit, eg. B. a monitoring circuit for monitoring the processing arrangement 104 , z. As a warning circuit, a dispatcher circuit, a communication circuit and the like to control.

According to various embodiments, the data fusion circuit 108 be configured by the processing arrangement 104 to model executed processes based on a process model and to predict the machining results based on the process model. The data merging circuit 108 can be configured to provide the results in real time, that is, as fast as possible, or in other words, with as little computational effort as possible, while modeling multiple processes (eg, manufacturing processes) in a complex and dynamic process environment.

According to various embodiments, the process control circuit 100 used in semiconductor manufacturing. In general, in addition to one or more internal control systems of the production tool, external automatic control is used to improve the stability of semiconductor manufacturing processes. This external automatic control is also referred to as pass-to-pass (R2R) control. The control loops almost always contain more than one production operation (eg pre-metrology, processing, post-metrology). Control can typically be provided by a central, factory-wide automatic system ( R2R System) which is operated 24/7.

According to various embodiments, a run-to-run strategy may include two steps. First, estimating an appropriate state (also referred to as state information) of the controlled processes using the process data of the previous materials. This data can be z. For example, in-line measurements (eg, a layer thickness, a critical dimension, etc.), derived quantities (such as the removal rate), and tool data. The state is chosen to reflect the characteristics of the process that needs to be tuned. Second, using state estimation to calculate the tuning parameter (s) for the next material.

• • um den Regelkreis im Fall eines ungültigen Zustands automatisch zu unterbrechen,
• • um zusätzliche Messungen automatisch auszulösen, um die Zustandsvalidität wiederherzustellen,
• • um die Anzahl der gemessenen Materialien (z. B. über das Metrologieabtastsystem) zu steuern,
• • um ein geeignetes Fertigungswerkzeug für das nächste zu bearbeitende Material automatisch auszuwählen (d. h., um ein Werkzeug auszuwählen, für das ein gültiger Zustand vorhanden ist),
• • um Materialien automatisch zu den Fertigungswerkzeugen zu senden, um das Ablaufen der entsprechenden Werkzeugzustände zu verhindern,
• • um die Rückkopplungsverstärkung des Regelkreises anzupassen, z. B. um die Rückkopplung zu vergrößern, falls der Zustand (teilweise) abgelaufen ist und/oder die ankommenden Messdaten signifikant neue Informationen enthalten.

According to various embodiments, the state estimation may be a major aspect of the control algorithm, ie, the process control circuit 100 and a method of controlling the processing arrangement 104 , In real life, the state information can be corrupted in various ways, e.g. By missing, delayed or erroneous measurements. It may be important to permanently validate the condition information to avoid yield loss and / or waste during machining or fabrication. Therefore, the processing arrangement 104 configured to also provide information about the accuracy of the state information, e.g. B. statistical error information such. A standard deviation of the estimation error. According to various embodiments, the validation routines may be part of the control algorithm. Without proper validation, the state estimates can be virtually worthless. According to various embodiments, the validity information (or in other words, one or more accuracy information) may be used for at least one of the following functions:

• • to automatically interrupt the control loop in the event of an invalid condition,
• • to automatically trigger additional measurements to restore state validity;
• • to control the number of measured materials (eg via the metrology scanning system),
• • to automatically select a suitable production tool for the next material to be machined (ie to select a tool for which a valid condition exists),
• • to send materials automatically to the production tools to prevent the expiration of the corresponding tool states,
• • to adjust the feedback gain of the control loop, eg. To increase the feedback if the state is (partially) expired and / or the incoming measurement data contains significantly new information.

• • der Zustand kann ungültig gemacht werden, falls der Zeitraum seit der letzten Zustandsaktualisierung irgendeinen vorgegebenen Schwellenwert übersteigt (Zeitabschaltung),
• • der Zustand kann ungültig gemacht werden, falls die Anzahl der bearbeiteten Materialien (z. B. Chargen, Wafer usw.) seit der letzten Zustandsaktualisierung irgendeinen vorgegebenen Schwellenwert übersteigt (Materialabschaltung),
• • die Zustandsaktualisierung kann unterdrückt werden, falls sich die Anzahl der einzelnen Messungen unter einem vorgegebenen Schwellenwert befindet,
• • der Zustand kann ungültig gemacht werden, falls die Anzahl der Materialien, die bearbeitet werden, aber noch nicht gemessen worden sind, einen vorgegebenen Schwellenwert übersteigt. Alternativ kann zu einer einfachen arithmetischen Summe der Anzahl der Materialien ebenso eine gewichtete Summe verwendet werden. Die gewichtete Summe hebt Materialien hervor, die zuletzt bearbeitet worden sind (eine unbeobachtete Drift),
• • In dem Fall, in dem ein EWMA-Filter (ein exponentiell gewichtetes Filter mit beweglichem Durchschnitt) für die Zustandsschätzung verwendet wird, kann die effektive Unterdrückung des Messrauschens aufgrund des Filters quantifiziert werden (Rauschverringerungsfaktor). Der Zustand kann nur gültig sein, falls die Rauschverringerung irgendeinen vorgegebenen Schwellenwert übersteigt.

Conventionally, simplistic validation criteria can be used for the surviving states of the processing arrangement 104 be used, such as. B .:

• The state can be invalidated if the period since the last state update exceeds any given threshold (timeout),
• • the state can be invalidated if the number of processed materials (eg batches, wafers, etc.) exceeds any given threshold since the last state update (material shutdown),
• • the status update can be suppressed if the number of individual measurements is below a predetermined threshold,
• • the state can be invalidated if the number of materials being processed but not yet measured exceeds a predetermined threshold. Alternatively, a weighted sum may also be used for a simple arithmetic sum of the number of materials. The weighted sum highlights materials that were last edited (an unobserved drift),
• In the case where an EWMA filter (an exponentially weighted moving average filter) is used for the state estimation, the effective suppression of the measurement noise due to the filter can be quantified (noise reduction factor). The condition can only be valid if the noise reduction exceeds any given threshold.

The thresholds for the simplistic validation criteria described above may be conventionally determined by a process engineer based on a "good feeling". The significance of the criteria is impaired by this highly subjective and error prone approach. In addition, the criteria are static and do not reflect the very dynamic nature of high mix, low volume manufacturing. Because of these limitations, the thresholds are traditionally conservative, causing premature interruptions in the control loop and other problems. In many cases, the validity information described above can not be implemented due to a lack of reliable validity information.

• • eine Werkzeugdrift (die Größe unterscheidet sich für verschiedene Produkte),
• • die Werkzeugwartung (eine zufällige und/oder systematische Zustandsverschiebung),
• • ein Gesamtausfall oder ein Mangel an Messwerten,
• • das Abtasten der Messwerte und/oder
• • zur Neige gehende Produkte.

The state estimation can be falsified by several mechanisms, such as: B .:

• • a tool drift (the size is different for different products),
• • tool maintenance (a random and / or systematic state shift),
• • a total failure or lack of measurements,
• • the sampling of the measured values and / or
• • Dwindling products.

Each of the simplistic validation criteria described above can cover only a partial aspect of the actual situation. A validation routine as described herein would be desirable, which can cover all spoofing mechanisms simultaneously.

• • eine feste Zeitabschaltung für eine Schätzung der Entfernungsgeschwindigkeit (der Zeitraum, in dem ein Schätzfehler über irgendeinen vorgegebenen Schwellenwert ansteigt, ist im hohen Grade variabel),
• • eine feste Materialabschaltung (die Anzahl der bearbeiteten Wafer, bevor der Schätzfehler irgendeinen vorgegebenen Schwellenwert verletzt, ist im hohen Grade variabel),
• • eine feste Anzahl einzelner Messungen pro Wafer, die für eine gültige Zustandsaktualisierung benötigt wird, (die durch jede Messung bereitgestellten Informationen unterscheiden sich in Abhängigkeit vom Ort der Messstelle beträchtlich),
• • eine feste Obergrenze für die Anzahl der bearbeiteten Wafer, die noch nicht gemessen worden sind, (verschiedene Produkte verschlechtern die Zustandsschätzung verschieden) und/oder
• • eine Untergrenze für den Rauschverringerungsfaktor (für die Berechnung des Rauschverringerungsfaktors wird angenommen, dass alle Produkte die Zustandsschätzung ähnlich verschlechtern; wobei außerdem fehlende oder verzögerte Messwerte vernachlässigt werden).

As an example, the removal rate of a chemical-mechanical planarization process may degrade for each processed wafer or processed product. The amount of deterioration may depend on the surface texture of the particular wafer or product being processed. The wafer or products can be processed in a virtually random order. Some batches can be measured immediately, some with a considerable time delay. The pass-to-pass control can be used to model the degradation of the removal rate and to tune the chemical mechanical planarization process, e.g. B. to set the distance time. In this complex situation, it is impossible for the process engineer to properly specify the following using a conventional pass-to-pass system

• A fixed timeout for an estimate of the range rate (the time period in which an estimation error rises above any given threshold is highly variable),
• A solid material shutdown (the number of wafers processed before the estimation error violates any given threshold is highly variable),
• A fixed number of individual measurements per wafer needed for a valid state update (the information provided by each measurement varies considerably depending on the location of the measurement site),
• A fixed upper limit for the number of processed wafers that have not yet been measured (different products worsen the state estimate) and / or
• A lower limit for the noise reduction factor (for the calculation of the noise reduction factor, it is assumed that all products similarly deteriorate the state estimation, besides neglecting missing or delayed measured values).

According to various embodiments, the state estimation and validation system may be constructed, as described below. It is stated that one skilled in the art can modify or rewrite the equations and / or algorithms described below without changing the general concept defined by the equations and / or algorithms. The indices used here can be z. Changed into other indices or replaced with other indices, any suitable programming language can be used to implement the algorithms. The variables in the form of scalars, vectors and / or matrices can also be reworded differently. Here, a general concept based on one or more examples is provided and described. Furthermore, detailed solutions for the preferred embodiments are also provided below, but are to be considered as exemplary, because the mathematical concept allows different formulations for the same equations.

In diesem Fall ist x ein Zustandsvektor (ausführlicher ist x_k+1 der zu schätzende unbekannte Zustandsvektor und ist x_k ein vorhergehender Zustandsvektor), ist A_k+1 eine Zustandsübergangsmatrix, ist ξ_k+1 ein Zufallsvektor, der das Prozessrauschen repräsentiert, (z. B. mit E[ξ] = 0; E[ξ²] = I), ist Q_k+1 eine Rauschskalierungsmatrix und ist B_k+1 ein Prozessdriftvektor, der deterministische Zustandsdrifts repräsentiert, und ist k eine Zählervariable. Gleichung (1) beschreibt die Entwicklung des Zustandsvektors x vom Zeitpunkt k bis zum Zeitpunkt k + 1 aufgrund eines einzigen Bearbeitungsschritts. Mit anderen Worten, es können mehrere durch die Bearbeitungsanordnung 104 ausgeführte Prozesse (die außerdem als Prozessereignisse bezeichnet werden) auf den Index k abgebildet werden, wobei der Index k außerdem die chronologische Reihenfolge der jeweiligen Prozesse repräsentieren kann. Es können verschiedene Parametermatrizen Q, B, A für jeden Zeitpunkt gemäß den Werkzeugen, Materialien usw., die in den Bearbeitungsschritt einbezogen sind, verwendet werden. Die Möglichkeit, verschiedene Matrizen/Vektoren für jeden Zeitschritt zu verwenden, kann für das präzise Modellieren der Auswirkung der verschiedenen Werkzeuge, Materialien usw. nützlich sein.According to various embodiments, a linear time varying state space model including random excitation to model the process, e.g. B. be used according to the following equation: $$x_{k+1}=A_{k+1}{x}_k+{\mathrm{<figure-callout\; id="1"\; label="Q\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">Q\; </figure-callout>}}_{k+}{\xi }_{k+1}+{\mathrm{<figure-callout\; id="1"\; label="B\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_{k+}$$

In this case, x is a state vector (x _{k + 1 is} the unknown state vector to be estimated, and x _{k is} a previous state vector), A _{k + 1 is} a state transition matrix, ξ _{k + 1 is} a random vector representing the process noise, ( For example, with E [ξ] = 0, E [ξ ² ] = I), Q _{k + 1 is} a noise scaling matrix, and B _{k + 1 is} a process drift vector representing deterministic state drifts, and is k a counter variable. Equation (1) describes the evolution of the state vector x from the moment k until time k + 1 due to a single processing step. In other words, several may pass through the processing arrangement 104 executed processes (also called process events) on the index k be imaged, where the index k In addition, it can represent the chronological order of the respective processes. There can be different parameter matrices Q . B . A at any time according to the tools, materials, etc. involved in the processing step. The ability to use different matrices / vectors for each time step can be useful for accurately modeling the impact of various tools, materials, and so on.

As illustrated above, according to various embodiments, the data merging circuit 108 be configured, the process data 105 based on a process model that includes one or more process model parameters for modeling known influences (eg, the process model parameters A . B ) and / or one or more process noise parameters for modeling unknown influences (eg, the process model parameters Q or N , as will be described in more detail below), to model.

The data merging circuit 108 can z. B. be configured, the process data 105 based on a process model (eg, a linear time-variant state space) containing at least one of the following process model parameters, to model: a state transition parameter (eg, the process model parameter A ), a process noise parameter (eg the process model parameter Q or N as described below) and / or a process drift parameter (eg, the process model parameter B ).

In diesem Fall ist x_k+1 ein Zustandsvektor, ist x_k ein vorhergehender Zustandsvektor, ist A_k+1 eine Zustandsübergangsmatrix, ist N_k+1 ein Prozessrauschvektor, ist B_k+1 ein Prozessdriftvektor und ist k eine Zählervariable. Mit N_k+1 = Q_k+1 ξ_k+1, wobei ξ_k+1 ein Zufallsvektor, der das Prozessrauschen repräsentiert, (z. B. mit E[ξ] = 0, E[ξ²] = I) ist und Q_k+1 eine Rauschskalierungsmatrix ist, gleicht die Gleichung der Gleichung (1).As an example, the data merging circuit 108 be configured, the process data 105 Similar to equation (1), model as follows: $$x_{k+1}=A_{k+1}{x}_k+N_{k+1}+{\mathrm{<figure-callout\; id="1"\; label="B\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_{k+}$$

In this case, x _{k + 1 is} a state vector, x _{k is} a previous state vector, A _{k + 1 is} a state transition matrix, N _{k + 1 is} a process noise vector, B _{k + 1 is} a process drift vector, and k a counter variable. With N _{k + 1} = Q _{k + 1} ξ _{k + 1} , where ξ _{k + 1 is} a random vector representing the process noise (eg, with E [ξ] = 0, E [ξ ² ] = I) and Q _{k + 1 is} a noise scaling matrix, the equation of equation (1) is similar.

In diesem Fall ist y der skalare Messwert, ist C der Beobachtungsvektor, der z. B. den Beitrag der Elemente des Zustandsvektors x zu dem skalaren Messwert y repräsentiert, ist η ein Zufallsvektor, der das Messrauschen repräsentiert, (z. B. in der Form E[η] = 0, E[η²] = 1), ist D ein Skalar, der zusätzliche deterministische Vorwärtsregelungseffekte modelliert, und ist k eine Zählervariable. Wie oben beschrieben worden ist, können verschiedene Parametermatrizen C, R, D für jeden Zeitschritt verwendet werden. Gemäß verschiedenen Ausführungsformen kann eine Anzahl von m entsprechenden unabhängigen Gleichungen (2) in dem Fall formuliert werden, in dem die Messung eine Anzahl von m (möglicherweise korrelierten) Messwerten bereitstellt. In diesem Fall können die m Messungen durch das sequentielle Ausführen der m Gleichungen (2) betrachtet werden.According to various embodiments, the corresponding measurement is modeled according to the process model, such. As follows: $$y_{k+1}=C_{k+1}{x}_{k+1}+R_{k+1}{\eta }_{k+1}+D_{k+1}$$

In this case is y the scalar reading, is C the observation vector, the z. For example, the contribution of the elements of the state vector x to the scalar reading y , η is a random vector representing the measurement noise (eg, in the form E [η] = 0, E [η ² ] = 1) D a scalar that models additional deterministic feed forward effects, and is k a counter variable. As described above, various parameter matrices can be used C . R . D be used for each time step. According to various embodiments, a number of m corresponding independent equations (2) may be formulated in the case where the measurement is a number of times m (possibly correlated) readings. In this case, the m Measurements by the sequential execution of the m Equations (2) are considered.

As has been illustrated above, the data merging circuit 108 be configured, the measured data 107 based on one or more measurement model parameters for modeling the known influences (eg the measurement model parameters C and or D ) and / or one or more measurement noise parameters for modeling the unknown influences (eg the measurement model parameters R and or M as will be described in more detail below). The data merging circuit 108 can z. B. be configured, the measurement data 107 based on a measurement model (the modeling of the measurement may correspond to the process model) that includes at least one of the following measurement model parameters: an observation parameter (the measurement model parameter C ), a measurement noise parameter (the measurement model parameter R or M ) and / or a measurement offset parameter (the measurement model parameter D ).

In diesem Fall ist y_k+1 ein skalarer Messwert oder ein Messvektor, ist C_k+1 ein Beobachtungsvektor oder eine Beobachtungsmatrix (der bzw. die den Beitrag der Elemente des Zustandsvektors x_k+1 zu y_k+1 beschreibt), ist M_k+1 ein Messrauschskalar oder ein Messrauschvektor, ist D_k+1 ein Messversatzskalar oder ein Messversatzvektor und ist k eine Zählervariable. Mit M_k+1 = R_k+1 η_k+1, wobei η_k+1 ein Zufallsvektor ist, der das Messrauschen repräsentiert, (z. B. mit E[η] = 0, E[η²] = I), und R_k+1 ein Rauschskalierungsvektor oder eine Rauschskalierungsmatrix ist, gleicht diese Gleichung der Gleichung (2).As an example, the data merging circuit 108 be configured, the measured data 107 Similar to equation (2), model as follows: $$y_{k+1}=C_{k+1}{x}_{k+1}+M_{k+1}+D_{k+1}$$

In this case, y _{k + 1 is} a scalar metric or measurement vector, C _{k + 1 is} an observation vector or observation matrix (which describes the contribution of elements of the state vector x _{k + 1} to y _{k + 1} ), M is M _{k + 1 is} a measurement noise scalar or measurement noise vector, D _{k + 1} is a measurement offset scalar or a measurement offset vector and is k a counter variable. With M _{k + 1} = R _{k + 1} η _{k + 1} , where η _{k + 1 is} a random vector representing the measurement noise (eg with E [η] = 0, E [η ² ] = I) , and Rk _{+ 1 is} a noise scaling vector or a noise scaling matrix, this equation is equal to equation (2).

According to various embodiments, not all elements of the matrices (A, N, Q, B, C, M, R) can be individually parameterized. One or more elements may be set to zeros. Large groups of elements can be assigned the same parameter. In this way, if desired, the effective number of parameters can be drastically reduced.

According to various embodiments, the production mass data (eg, exemplary process data 105 and measurement data 107 of the respective process to be modeled) over a given period of time. A statistical procedure (eg, a maximum likelihood estimate, as will be described in more detail below) may be used to determine the parameters N, Q, M, R, B, D, C, A of the process model described above, e.g. For example, according to equations (1), (2) or (1 *), (2 *), or any other suitable process model based on similar equations.

According to various embodiments, a Kalman filter may be used to determine the condition x and to estimate the corresponding estimation error covariance matrix P (the estimation error covariance matrix P can predict the accuracy of the state x describe about the process model). The filter can be parameterized with the matrices, the z. B. determined by the statistical procedure described above. According to various embodiments, the Kalman filter may be divided into two sets of equations Therefore, updating and restoring the previous state of the state information can be performed independently of each other.

$$\begin{array}{c}{\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)\\ P_{k+1}=\left(I-K_{k+1}{C}_{k+1}\right)P_k\\ K_{k+1}=P_k{C}_{k+1}^T{\left(C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T\right)}^{-1}\end{array}.$$

According to various embodiments, the Kalman filter may e.g. Based on equations (1), (2), as follows, into two sets of equations (3), (4): $$\begin{array}{c}{\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+{\mathrm{<figure-callout\; id="1"\; label="B\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_{k+}.\\ P_{k+1}=A_{k+1}{P}_k{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T.\end{array}$$

$$\begin{array}{c}{\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)\\ P_{k+1}=\left(I-K_{k+1}{C}_{k+1}\right)P_k\\ K_{k+1}=P_k{C}_{k+1}^T{\left(C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T\right)}^{-1}\end{array},$$

In the same way, the Kalman filter can be based on e.g. For example, the equations (1 *), (2 *) or similar equations of another suitable process model may be divided into two sets of equations. Both sets of equations (3), (4) can be performed completely independently of each other. Equation (3) may be executed after each process or maintenance step (also referred to as "process update" or just "update"). These steps increase the estimation error and therefore reduce the accuracy of the state information as a new unknown process noise shifts the observed state. Equation (4) may be performed after one or more measurement steps (which is also referred to as "metrology update" or only as "restore previous state"). Considering the measurement data reduces the estimation error and therefore increases the accuracy of the state information because it provides new information about the state. In the case where the metrology data (also referred to as the measurement data) consists of more than one measurement, the metrology update ( 4 ) for each value separately. The order of ( 3 ) and ( 4 ) accurately reflects the accumulation and degradation of the process information (eg, the decrease and increase of the estimation error) as long as the process model parameters N, Q, M, R, B, D, C, A are real life processes fit, the z. B. by the processing arrangement 104 be executed. According to various embodiments, the process model, which includes independent prior art update and recovery as described herein, provides a unified framework for simultaneously modeling and monitoring all relevant state corruption mechanisms, which will also be described in more detail below.

In diesem Fall können eine oder mehrere Zustandskombinationen von Interesse durch den Zustandsextraktorvektor C definiert werden. Gemäß verschiedenen Ausführungsformen kann die Standardabweichung σ_ERR des Schätzfehlers verwendet werden, um die Zustandskombination zu validieren. Die Standardabweichung σ_ERR des Schätzfehlers weist die gleiche physikalische Einheit wie die beobachtete Zustandskombination auf. Gemäß verschiedenen Ausführungsformen kann ein geeigneter Schwellenwert in Kombination mit der Standardabweichung σ_ERR des Schätzfehlers verwendet werden, um den Regelkreis unterbrechen, um zusätzliche Messungen auszulösen usw., wie oben beschrieben worden ist. Die Verwendung der Fehlerkovarianzmatrix P für die Zustandsvalidierung kann hilfreich und ein Hauptunterschied zu den üblicherweise verwendeten Durchlauf-zu-Durchlauf-Kalman-Filter-Anwendungen sein.According to various embodiments, the error covariance matrix P _{k + 1} may be used to calculate the error variance or, in other words, to provide the accuracy information of the state information. The error variance can be determined based on the following equation or similar formulations: $${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2=\overline{C}{P}_{k+1}{\overline{C}}^T$$

In this case, one or more state combinations of interest may be provided by the state extractor vector C To be defined. According to various embodiments, the standard deviation σ _{ERR of} the estimation error may be used to validate the state combination. The standard deviation σ _{ERR of} the estimation error has the same physical unit as the observed state combination. According to various embodiments, an appropriate threshold in combination with the standard deviation σ _{ERR of} the estimation error may be used to interrupt the loop to trigger additional measurements, etc., as described above. The use of the error covariance matrix P for state validation can be helpful and a major difference to the commonly used pass-to-pass Kalman filtering applications.

According to various embodiments, the state vector x and the error covariance matrix P may be automatically expanded when the data for new products, a new tool, etc. are acquired. Therefore, the run-to-run controller (eg, the process control circuit 100 ) be able to adapt to the ever-changing environment of semiconductor production or any other processing environment.

2A illustrates the operation of the data fusion circuit 108 or, in other words, the operation of the process control circuit 100 based on a graphical representation according to various embodiments. The graph shows the accuracy of the state information 202y (eg, based on a standard deviation σ _{ERR of} the error or based on a normalized renewal of the ratio (which can be understood, for example, as a normalized prediction error), as will be described in more detail below) for a single state model over a time index . k , 202x. The corresponding threshold 202t can z. B. can be used to interrupt the control loop to trigger additional measurements, etc., as described above. As illustrated, the time index is used k = 150 up to the time index k = 300 a total failure of the measurements assumed. The accuracy of the state information (eg, the standard deviation of the error) exceeds the threshold just after the start and in the period of total failure of the measurements 202t , After the total failure of the measurements, the accuracy of the state information decreases and falls rapidly below the threshold 202t , Both effects (settling and total failure of the measurements) may be monitored by the same threshold according to various embodiments.

2 B illustrates the operation of the data fusion circuit 108 or, in other words, the operation of the process control circuit 100 based on a graphical representation according to various embodiments. The graph illustrates the accuracy of the state information 202y (eg, based on a standard deviation σ _{ERR of} the error or based on a normalized renewal of the ratio, as described in more detail below) for four products processed at different frequencies. P1 to P3 are z. B. high-speed products while P4 a deep running product. The magnitude of the estimation error (or any other suitable value to determine the accuracy of the state information) differs considerably due to the dissimilar processing frequencies. According to various embodiments, the data fusion circuit 108 as described here, model this effect properly. In addition, the estimation error increases when the material is processed without tracking measurements. This effect is also caused by the data fusion circuit 108 covered. According to various embodiments, the predicted estimation error is accumulated until a new measurement arrives. Due to the metrology update (which is also referred to as restoring the previous state), the estimation error drops and then accumulates again.

• sie kann eine Zustandsschätzung bereitstellen, die bezüglich der verfügbaren Daten in Anbetracht dessen optimal ist, dass:
  • • Daten bei der Fertigung in geringem Volumen und hoher Mischung knapp, unregelmäßig und wertvoll sind,
  • • die Rückkopplungsverstärkung an die Datensituation und die Wafer in Bearbeitung (WIP) angepasst werden kann,
  • • die Schätzungen für jeden Zeitpunkt optimal sind,
  • • eine Zustandsschätzung selbst in ungünstigen Situationen (z. B. fehlende Messwerte, fehlende Abtastung usw.) optimal ist;
• ein Schätzfehler in Anbetracht dessen bereitgestellt (z. B. berechnet) werden kann, dass:
  • • eine Empfehlung unterdrückt werden kann, falls die Genauigkeit einer Zustandsschätzung unzureichend ist,
  • • eine permanente Modellvalidierung verwirklicht ist, es z. B. geprüft werden kann, ob ankommende Daten konsistent modelliert werden und/oder ob das für die Vorhersage verwendete interne Modell richtig bereitgestellt ist und dem Prozess im wirklichen Leben entspricht,
• ein universeller Rahmen für eine einfache und eine komplexe Steueraufgabe bereitgestellt wird; und
• eine modellbasierte Herangehensweise bereitgestellt wird, die sich ohne die Notwendigkeit des Anpassens der Schätzalgorithmen selbst nur auf die richtige Modellierung des Prozesses konzentriert.

According to various embodiments, the process control circuit described herein may be used 100 provide at least one or all of the following:

• it can provide a state estimate that is optimal in terms of available data given that:
  • • Low-volume and high-mix production data is scarce, irregular and valuable,
  • • the feedback gain can be adapted to the data situation and the wafer in progress (WIP),
  • • the estimates are optimal for each time point,
  • A state estimation is optimal even in unfavorable situations (eg missing readings, missing sample, etc.);
• an estimation error can be provided (eg calculated) considering that:
  • • a recommendation can be suppressed if the accuracy of a state estimation is insufficient,
  • • a permanent model validation is realized, it z. For example, it is possible to check whether incoming data is consistently modeled and / or whether the internal model used for the prediction is properly provided and corresponds to the process in real life.
• providing a universal framework for a simple and a complex tax task; and
• a model-based approach is provided which, without the need to adjust the estimation algorithms themselves, focuses only on the proper modeling of the process.

According to various embodiments, the data fusion circuit 108 a component of a run-to-run system (eg in semiconductor manufacturing). According to various embodiments, the data fusion circuit 108 as a component of one (e.g. factory-wide) automatic pass-to-pass control system. Such systems may be present in any modern semiconductor factory. The use of the data fusion circuit 108 however, can not be limited to central (factory-wide) systems; it could also be used for a local system as well. The task of R2R Systems may be to modify (eg, tune or adjust) the process parameters of critical production processes. The R2R System can control many production processes simultaneously and can keep the state information (eg state estimation) for all controlled processes, typically in a database. At each relevant production event (eg a process, a measurement, a maintenance, etc.) data is sent to the R2R System be sent or, in other words, data through the system R2R System recorded. The R2R System can be configured to select the state that corresponds to the current production step, adding the new (collected) data to it.

According to various embodiments, when the production event degrades the state estimation (eg, due to a machining event, a maintenance event, etc.), data is added by equation (3), which is also referred to as a process update. When the production event has generated the data (eg, in the case of a metrology event) that improves the state estimate, the data is added by equation (4), which is also referred to as a metrology update.

• • Wenn die Standardabweichung σ_ERR des Schätzfehlers einen vorgegebenen Schwellenwert, L_REC, übersteigt, können die R2R-Systeme keine Abstimmparameter an die Bearbeitungsanordnung 104, z. B. an ein Produktionswerkzeug, senden. Der automatische Regelkreis (z. B. die Steuerung der Bearbeitungsanordnung 104) kann unterbrochen werden, um einen Ausbeuteverlust und/oder Ausschuss zu vermeiden. Gemäß verschiedenen Ausführungsformen kann die Datenverschmelzungsschaltung 108, die wenigstens eine Validierungsroutine enthält, sicherstellen, dass die automatische Durchlauf-zu-Durchlauf-Steuerung (z. B. die Steuerung der Bearbeitungsanordnung 104) nur ausgeführt wird, wenn eine ausreichend zuverlässige Zustandsschätzung verfügbar ist.
• • Die ankommenden Messdaten y_k+1 können z. B. durch das Berechnen der normierten Erneuerung des Verhältnisses, v₀, z. B. gemäß Folgendem: $${\mathrm{<figure-callout\; id="0"\; label="v"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">v</figure-callout>}}_0=\frac{y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}}{\sqrt{C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T}}$$
  
  validiert werden. In diesem Fall können die gültigen Messungen durch -Ly ≤ v₀ ≤ Ly, z. B. mit 1 ≤ Ly ≤ 5 oder 2 ≤ Ly ≤ 4 oder irgendeinem anderen geeigneten Intervall als ein Kriterium, charakterisiert werden. Die normierte Erneuerung v₀ des Verhältnisses ist ein universelles Maß, das von dem Verwendungsfall und dem Übergangsverhalten (dem Einschwingen) des Schätzalgorithmus unabhängig ist. Falls die normierte Erneuerung v₀ des Verhältnisses Ly übersteigt, sind mehrere Reaktionen möglich, wie z.B.:
  • ■ es kann eine Zustandsaktualisierung übersprungen werden, weil y_k+1 als fehlerhaft betrachtet wird, und/oder
  • ■ es kann ein Prozessingenieur (z. B. über E-Mail, SMS oder irgendein anderes geeignetes Signal oder irgendeine andere geeignete Nachricht) informiert werden, weil das Modell und die realen Daten nicht mehr zusammenpassen; dies könnte eine Angabe sein, dass sich der Prozess geändert hat und das Prozessmodell neu eingestellt werden muss.
• • Das Metrologieabtastsystem kann basierend auf den Genauigkeitsinformationen, z. B. basierend auf der Standardabweichung σ_ERR des Schätzfehlers, dynamisch gesteuert werden. Für hohe Werte der Standardabweichung σ_ERR des Schätzfehlers kann die Abtastrate verringert werden (d. h., es können mehr Materialien gemessen werden); für geringe Werte der Standardabweichung σ_ERR des Schätzfehlers kann die Abtastrate vergrößert werden (es können z. B. weniger Materialien gemessen werden). Dafür kann gemäß verschiedenen Ausführungsformen eine Schnittstelle und ein Datenaustausch zwischen dem Abtasten und dem R2R-System bereitgestellt werden.
• • Wenn die Standardabweichung σ_ERR des Schätzfehlers einen vorgegebenen Schwellenwert, L_MET, übersteigt, können zusätzliche Messungen ausgelöst werden, z. B. über ein Echtzeit-Abfertigungssystem (RTD-System), um die Genauigkeit der Zustandsschätzung zu verbessern. Dafür kann gemäß verschiedenen Ausführungsformen eine Schnittstelle und ein Datenaustausch zwischen dem RTD- und dem R2R-System bereitgestellt werden.
• • Gemäß verschiedenen Ausführungsformen kann das Echtzeit-Abfertigungssystem konfiguriert sein, basierend auf der Standardabweichung σ_ERR des Schätzfehlers ein neues Produktionswerkzeug für ein spezifisches Material auszuwählen:
  • ■ wobei nur Werkzeuge ausgewählt werden können, für die die Standardabweichung des Schätzfehlers σ_ERR < L_REC ist; dies stellt sicher, dass für alle ausgewählten Werkzeuge ein gültiger Zustand zur Berechnung der Abstimmparameter verfügbar ist, und/oder
  • ■ aus den ausgewählten Werkzeugen das Werkzeug mit der höchsten Standardabweichung σ_ERR des Schätzfehlers gewählt werden kann; dies kann eine weitere Verschlechterung/einen weiteren Ablauf der entsprechenden Zustandsschätzung verhindern.

If material (eg, a wafer, a batch, a lot, etc.) arrives at a controlled production process, a request to the R2R System are sent. The R2R System may use the state information and / or the accuracy information to predict the outcome of the process. The prediction may be used to calculate one or more tuning parameters that may improve the result according to a predetermined optimization criterion. The accuracy information (eg, the standard deviation σ _{ERR of} the estimation error given by the equation (5)) may be used as follows:

• • If the standard deviation σ _{ERR of} the estimation error exceeds a predetermined threshold, L _REC , the R2R systems can not apply tuning parameters to the processing arrangement 104 , z. To a production tool. The automatic control loop (eg the control of the processing arrangement 104 ) can be interrupted to avoid loss of yield and / or rejects. According to various embodiments, the data fusion circuit 108 containing at least one validation routine, ensure that the automatic run-to-pass control (eg, the control of the processing arrangement 104 ) is executed only when a sufficiently reliable state estimate is available.
• • The incoming measurement data y _{k + 1} can, for. By calculating the normalized renewal of the ratio, v ₀ , z. B. according to the following: $${\mathrm{<figure-callout\; id="0"\; label="v"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">v</figure-callout>}}_0=\frac{y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}}{\sqrt{C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T}}$$
  
  be validated. In this case, the valid measurements can be given by -Ly ≤ v ₀ ≤ Ly, z. With 1 ≦ Ly ≦ 5 or 2 ≦ Ly ≦ 4 or any other suitable interval as a criterion. The normalized renewal v _{0 of} the ratio is a universal measure that is independent of the use case and the transient behavior (the settling) of the estimation algorithm. If the normalized renewal exceeds v _{0 of} the ratio Ly, several reactions are possible, such as:
  • ■ a status update can be skipped because y _{k + 1 is} considered faulty, and / or
  • ■ a process engineer can be informed (eg via e-mail, SMS or any other suitable signal or any other suitable message) because the model and the real data no longer match; this could be an indication that the process has changed and the process model needs to be reset.
• • The metrology scanning system can be based on the accuracy information, e.g. Based on the standard deviation σ _{ERR of} the estimation error, dynamically controlled. For high values of the standard deviation σ _{ERR of} the estimation error, the sampling rate can be reduced (ie, more materials can be measured); for low values of the standard deviation σ _{ERR of} the estimation error, the sampling rate can be increased (for example, fewer materials can be measured). For that can According to various embodiments, an interface and a data exchange between the scanning and the R2R System be provided.
• • If the standard deviation σ _{ERR of} the estimation error exceeds a predetermined threshold, L _MET , additional measurements may be triggered, eg. Via a real-time dispatch system ( RTD System) to improve the accuracy of the state estimation. For this, according to various embodiments, an interface and a data exchange between the RTD - and the R2R System be provided.
• According to various embodiments, the real-time dispatch system may be configured to select a new production tool for a specific material based on the standard deviation σ _{ERR of} the estimation error:
  • ■ where only tools can be selected for which the standard deviation of the estimation error is σ _ERR <L _REC ; this ensures that a valid state for calculating the tuning parameters is available for all selected tools, and / or
  • ■ the tool with the highest standard deviation σ _{ERR of} the estimation error can be selected from the selected tools; this can prevent a further deterioration / further course of the corresponding state estimation.

According to various embodiments, based on the accuracy information, one or more monitoring processes may be started and / or one or more warnings may be sent or displayed. There may be more than one threshold to start more than one different process according to the respective threshold in the case where the accuracy of the state information exceeds that respective threshold.

According to various embodiments, the data fusion circuit 108 or the process control circuit 100 may be a major component of a discrete process control system (e.g., batch processes) in any other industries. The embodiments are described herein with respect to semiconductor processing. The use of the data fusion circuit 108 or the process control circuit 100 however, it can not be limited to the semiconductor industry. It can be used to control any discrete process that can be described by equations (1) and (2) or similar equations. The data merging circuit 108 provides a general condition validation means which may be part of a control system.

3A shows a schematic diagram of the operation of a process control circuit 100 that is used to process a process 300 a processing arrangement 104 and a method of controlling a processing arrangement 104 according to various embodiments.

In analogy to the embodiments described above, the process control circuit 100 be configured, the process data 105d and the measurement data 107d a processing arrangement 104 to capture, with the process data 105d multiple process event data according to multiple process events 105e and the measurement data may include multiple measurement event data in accordance with multiple measurement events 107e can contain. The process control circuit 100 may be further configured, estimated state information 302 (see the reference number X ), where the estimated state information X an estimated status of the processing arrangement 104 and error information representing an estimation error σ (see reference numeral .DELTA.X ) of the estimated state information X represent, provide. The process control circuit 100 may be configured, the estimated state information X by updating 300u (eg, by performing a process update according to Equation 3) and restoring the previous state 300d (eg, by performing a metrology update or an earlier state restore according to Equation 4) of the previous state information based on the acquired process event data 105d and the acquired measurement event data 107d wherein the estimation error σ (calculated, for example, according to Equation 5) is provided by updating 300u (eg, considering the process event data 105d ) and wherein the estimation error σ is restored by restoring 300d of the previous state (eg, in view of the measurement event data 107d ) is reduced. The process control circuit 100 can be configured, the processing arrangement 104 based on the estimated state information X and / or the estimation error σ 111.

Based on the estimation error σ, at least one additional circuit can be controlled; B. a real-time dispatcher to be controlled 311 , the scanning system can be controlled 311b , a tuning parameter can be determined 311c or it can be the incoming measurement event data 105d be validated 311d , According to various embodiments, at least one additional circuit may be provided by sending a message via a communication circuit 312 (which, for example, contains any suitable circuitry to send at least one message).

According to various embodiments, the method of controlling the processing arrangement 100 Contains: Capturing process data 105 and measurement data 107 the processing arrangement, wherein the process data comprises a plurality of process event data 105d according to several process events 105e contain and the measurement data multiple measurement event data 107d according to several measurement events 107e contain; Providing status information X that have a status of the processing arrangement 104 represent and accuracy information .DELTA.X providing an accuracy of the state information X represent; To update 300u the state information X (eg, updating previous state information) based on the captured process event data 105d , which improves the accuracy of the state information X is reduced and independent of updating 300u the state information X Restore 300d the previous state of state information X (eg, restoring the previous state of the previous state information) based on the acquired measurement event data 107d , which improves the accuracy of the state information X is increased, and controlling the processing arrangement 104 based on the state information X and / or based on the accuracy information .DELTA.X ,

According to various embodiments, the process control circuit 100 or the data fusion circuit 108 as a main component of a surveillance system (eg in the semiconductor industry), as in 3B can be used in analogy to the embodiments described above.

In semiconductor manufacturing many processes can be monitored. It can, for. B. a statistical process control ( SPC ) or a fault detection and classification ( FDC ) can be used for monitoring. According to various embodiments, the data merge circuit described herein 108 also be used as an improved monitoring system. According to various embodiments, monitoring may include: selecting a suitable process model (eg, based on equation (1)) and a suitable measurement model (eg, on equation (2)) for size, y which needs to be monitored, and tuning the process model parameters by a statistical procedure using previous (eg, historical) size data y (eg, about maximum likelihood estimation, as described in more detail below); Implementing equations (3) and (4) in the monitoring system (eg, via a data merging circuit described herein 108 ) for calculating a state estimate x _k , P _k and computation (see reference numeral 313 ) of the normalized renewal, v ₀ (eg based on equation (6)) or any other error information of state estimation. When new data y _{k + 1} arrives, the monitoring may validate (see the reference numeral 315 ) of the new data 107d and / or validating (see reference numeral 315 ) of the process model. The monitored process can, for. B., that is, it behaves normally in the case where -Ly ≦ v ₀ ≦ Ly with 1 ≦ Ly ≦ 5 or 2 ≦ Ly ≦ 4.

According to various embodiments, the process control circuit 100 or the data fusion circuit 108 in a monitoring system generally used to monitor discrete processes, e.g. B. with the equations (1) and (2) or similar equations can be modeled.

According to various embodiments, the process control circuit 100 or the data fusion circuit 108 may be used as part of a device (eg, a controller) that tunes and / or monitors specific semiconductor processes (eg, chemical mechanical planarization, lithography, etching, deposition, etc.). According to various embodiments, the process control circuit 100 or the data fusion circuit 108 may be used as part of a device (eg, a controller) that tunes and / or monitors specific processes in any other industry.

According to various embodiments, a Kalman filter having a corresponding error covariance matrix P used to validate the estimated state of a control and / or monitoring system. According to various embodiments, a Kalman filter having a corresponding error covariance matrix P used to validate the measurement data collected by a control and / or monitoring system.

Such as In 3A is illustrated, the process control circuit 100 both functions, a control function and / or a monitoring function included. According to various embodiments, the one or more functions of the process control circuit 100 or the data fusion circuit 108 be implemented as a software module, e.g. For use in a run-to-run control system, e.g. B. Control manufacturing processes on a 24/7 basis.

The process control circuit 100 or the data fusion circuit 108 enables semiconductor production processes to be controlled and monitored much more accurately than based on conventional algorithms, as described herein. An advantage may be the accurate validation of the current state of the control / monitoring system even for very complex production scenarios (eg, a high mix, a low volume, a significant time lag between the process and the measurement, a metrology scan, etc.).

In a high-mix, low-volume production, data can be scarce, erratic, and valuable. The process control circuit 100 or the data fusion circuit 108 as described herein maximizes the available data, improves the quality of pass-to-pass control, increases the stability of production and / or avoids yield losses and rejects.

According to various embodiments, yield loss and / or rejection due to invalid state estimates may be prevented using the accuracy information. The loop can only be interrupted if necessary because any unnecessary disruption would generate additional production costs in the form of pre-sent wafers, additional workload for the production personnel (eg in case of manual intervention) and / or loss of time. In addition, costs can also be saved.

According to various embodiments, one or more aspects of the process control circuit 100 or the data fusion circuit 108 in software, eg Based on the MATLAB® software or any other suitable software.

According to various embodiments, the recursive estimation and prediction algorithms may be adapted to production scenarios with delayed and permutated measurements. In order to improve the stability of semiconductor manufacturing processes or other manufacturing processes, external automatic control may be used in addition to the internal control of the production tool. This external control can be used as a run-to-run control ( R2R Control), as described above. According to various embodiments, a run-to-run strategy may include at least two steps: first, estimating an appropriate state of the controlled processes using the process data of the previous materials, and second computing a tuning parameter (tuning parameters) for the next material (the next Batch, the next wafer, etc.) based on the estimate.

In general, state estimation can be an important aspect of the control algorithm. It may be advantageous to use a Kalman filter-based state estimation algorithm as described herein. The Kalman filter may be refreshed (or, in other words, updated) at least twice for a material. The first refresh (or first update) may be performed (eg, directly) after editing the material (ie, the process update), while the second refresh (or second update) may be performed if the corresponding metrology data is available (ie, a previous state restore or a metrology update). There is often a considerable time delay (eg, several hours or days) between processing and measuring, and therefore, one or more other manufacturing operations between updating and restoring the previous state for a material (eg, a process event and a corresponding measurement event of a material) can be performed. In the meantime, other materials can be edited and / or measured. These other materials may overhaul the previously processed materials, that is, the materials arrive in a randomly permuted order in the metrology operation, see e.g. B. 3A , To use the new information as soon as possible, any update can be performed immediately. However, the Kalman filter is a recursive estimation algorithm that may require that the materials be updated in a strict order, ie, in order PU ( k ) MU ( k ) PU ( k + 1), MU ( k + 1), PU ( k + 1), MU ( k + 1) etc., where PU ( k ) the process update for a material k (or an event k ) and MU ( k ) the metrology update for the material k (or the event k ) designated. The index defined by the respective process event k can be the counted variable for the process model, as described here.

4A Figure 12 illustrates a schematic diagram of a natural update sequence due to a manufacturing process. In other words, the processing arrangement 104 can be operated, resulting in the illustrated order for the process updates PU and the measurement updates MU for a first processed object (with the index 1 ), a second processed object (with the index 2 ), a third processed object (with the index 3 ), a fourth edited object (with the index 4 ) and a fifth processed object (with the index 5 ) as an example. Because, according to various embodiments, updating and restoring the previous state of the data fusion circuit 108 can be performed independently of one another, as described above, the illustrated natural order for the process updates PU and the measurement updates MU by the operation of the processing arrangement 104 is defined as the update order of the filter.

However, according to various embodiments, in a schematic diagram, the natural update sequence may be modified into a (sorted) update sequence due to the manufacturing process that is appropriate for a recursive filter, e.g. B. for a Kalman filter, may be advantageous, as in 4B is illustrated. Without transforming the natural sequence into the sorted sequence, the Kalman filter can not work optimally. Other recursive estimation algorithms (such as the Wiener-Kolmogorow filter) may be used in a similar manner.

As in the 4A and 4B is illustrated, the index (eg, the index k ) of a material by the process event, wherein the measurement event is correlated with the respective process event to provide the process and measurement data for each of the processed materials.

According to various embodiments, a process control circuit similar to the process control circuit described above 100 based on recursive estimation algorithms to estimate the state information. In this case, the process control circuit may include: a data acquisition circuit that is configured, the process data 105 and the measurement data 107 a processing arrangement to capture, the process data 105 multiple process event data 105d according to multiple process events, and the measurement data includes multiple measurement event data 107d contained according to several measurement events; a recursive process control circuit configured to update current state information 402c based on the process data 105 a current process event and a sequence 400Q recalculate previous process events recursively and the current state information 402c when capturing the measurement event data 107d representing at least one previous process event of the sequence 400Q Previous process events correspond to recalculate, and a control circuit configured, the processing arrangement based on the current state information 402c to control.

According to various embodiments, all materials with outstanding metrology data may be queued 400Q be collected, such. In 4C is illustrated in a schematic view. The queue 400Q can according to the processing time (eg by the index k is defined, where k a natural number) of the materials can be sorted in ascending order, see 4B , However, one or more measurement event data may be missing, that for the data merge circuit 108 may be configured to cope with the missing measurement event data, as described herein.

According to various embodiments, for the process update 300u a new entry in the queue every time a new material is edited 400Q be generated. The position of the item in the queue 400Q can z. B. be determined by the processing time (see 4C ). According to various embodiments, for the metrology update 300d every time metrology data is queued for a material 400Q arrive, the corresponding queue element ( 1 to n . n is a natural number) to be marked by a flag. According to various embodiments, a base state 402b being held. The base state may be the state of the Kalman filter (or any other recursive filter or recursive state estimation algorithm) just before the first material (with the index 1 ) in the queue 400Q is looked at. After every process update 300u or any metrology update 300d For example, the iterations of the Kalman filter are performed on all materials (eg, 1 through n, n is a natural number) in the queue. The iteration may be with the base state 402b be initialized. Illustratively, the iteration moves along the queue 400Q (For example, the materials are updated according to their processing time, ie the "old" materials are updated first). For each material, a process update is performed. The metrology update will only be executed if metrology data for the particular material Are available. The result of the iteration is considered a current state 402c saved. This can be z. B. quick access to the current state 402c without the need to queue 400Q recalculate, allow.

As in 4D In a schematic diagram, the materials may be queued 400Q be removed 400r if the metrology data is available and there is no legacy material with missing metrology data.

As in 4E In a schematic diagram, according to various embodiments, the length of the queue 400Q be limited. This may be advantageous in some applications because it prevents excessive computational load, memory usage, and update time. The aspect of collecting the material with outstanding metrology data in a queue 400Q and updating the state by iterating along the queue can be applied to any recursive algorithm of a controller.

According to various embodiments, a recursive estimation algorithm (eg, a Kalman filter) may be used to estimate the state information for a pass-to-pass control system as described herein. According to various embodiments, the state information may be the base state 402b (eg, a state vector and an error covariance matrix), the queue 400Q and the current state 402c contain. For each element of the queue 400Q can provide contextual logistical information (which also acts as the process data 105 be designated). This logistical context information may be related to the incoming metrology data (which, in addition, is the measurement data 107 be designated) for the corresponding queue element.

For each relevant production event (eg, a process, a measurement, a maintenance, etc.), data may be sent to the R2R System are sent. The R2R System can select the state that corresponds to the current production event, adding new data to it. If the production step corresponds to a process update, a new element of the queue is created and the entire queue is subsequently recalculated. If the production event matches a metrology update, the metrology data is assigned to the appropriate element of the queue, and then the entire queue is recalculated. If material arrives at a controlled production event, a request can be sent to the R2R system. The R2R System can use the state information to predict the outcome of the process and calculate the tuning parameters.

As in 4A is illustrated, the data fusion circuit 108 be configured to update the state information for each of the acquired process event data and for each of the acquired process metrics in a chronological order defined by the process events 300u and restore their former condition 300d , where the trade fair events are mapped to the process events. In this case, z. B. the data merge circuit 108 be configured to update the state information for each of the plurality of process events in acquiring the corresponding process event data and for each of the plurality of measurement events in acquiring the corresponding measurement event data 300u ,

Furthermore, the data merging circuit 108 configured to collect a plurality of process event data of the respective process events and to update the state information in consideration of the collected plurality of process event data upon acquiring the measurement event data of a subsequent trade event and to restore the previous state of the state information based on the acquired measurement event data.

Alternatively, the order for updating 300u and restoring 300d the previous state of the state information are sorted as in 4B is illustrated and / or queued 400Q be processed, as in the 4C to 4E is illustrated. In this case, z. The data acquisition circuit 102 further configured to correlate the measurement events with the corresponding process events (eg mapping the measurement events to the corresponding process events). According to various embodiments, for each material (eg, for each with the processing arrangement 104 machined workpiece) provides at least one (eg, one or more) tuples based on the process event data and the measurement event data corresponding to the process event, the tuple illustratively representing a product, e.g. As a wafer can be assigned. Furthermore, the data merging circuit 108 be configured, the current state information 402c based on the process event data of a current process event and a sequence of previous process events and / or measurement events to calculate. The sequence of the previous process events and / or measurement events can be determined by the queue 400Q be defined. The data merging circuit 108 can be configured, the current state information 402c based on the process event data of a current process event and a sequence of previous process events and the current state information 402c upon recalculating measurement event data corresponding to at least one previous process event of the sequence of previous process event data. The sequence of previous process events may be a limited number of process events, starting from an initial process event containing the base state information 402b defines, and ending with the current process event, the current state information 402c defined, included. In this case, the initial process event may be selected to be a first item in the sequence of the previous process events for which no corresponding measurement event data is available and for which the corresponding measurement event data is to be considered. According to various embodiments, the length of the queue may be 400Q be reduced to a desired maximum number of elements.

According to various embodiments, a method is provided for coping with complex ambiguities in run-to-run Kalman filter models. As described above, the state estimation may be an important part of a control algorithm that may be used in a process control circuit. According to various embodiments, the control algorithm may be based on a Kalman filter, see e.g. For example, the equations (1) to (6) have been described above.

The state information of the process (see, for example, equations (1) and (2)) may be estimated by two sets of equations (eg, sets of equations (3) and (4)) that are performed completely independently of each other , After each process event or maintenance event, a process update may occur 300u (for example, see the first set of equations (3)). Further, metrology data may be added to the condition through a metrology update 300d (see eg the second set of equations (4)).

• ■ eine sehr große Anzahl von Zuständen (50-1000);
• ■ neue Zustände müssen automatisch hinzugefügt und initialisiert werden;
• ■ Zustände entwickeln sich über längere Zeiträume (Jahre);
• ■ irgendein Zustand wird (im Wesentlichen) niemals aktualisiert;
• ■ eine Modellmehrdeutigkeit.

The sequence of sets of equations (3), (4) accurately reflects the accumulation and degradation of the process information (the decrease and increase of the estimation error) as long as the process model parameters Q, R, B, D, C, A are Process fit in real life. However, the practical application of the Kalman filter for pass-to-pass control can be adjusted to accommodate one or more of the following:

• ■ a very large number of states ( 50 - 1000 );
• ■ new states must be added and initialized automatically;
• ■ Conditions develop over longer periods of time (years);
• ■ any state is (essentially) never updated;
• ■ a model ambiguity.

According to various embodiments, a new state (may be new states) at runtime may be increased by increasing the state information X and the error covariance matrix P are added. However, it can at least be helpful to initialize the new state correctly. According to various embodiments, this can be done in the data fusion circuit 108 filters used to interpret the initial values as an earlier knowledge of the process, which in turn directly affects the result of the state estimation.

However, reliable prior knowledge may often not be available in practice. This may be the case, in particular, for an automatic system that operates 24/7. According to various embodiments, the filter that is used may be the processing assembly 104 be configured in such a way that the initial values do not contain information. This can be done by initializing the diagonal elements of the error covariance matrix P , which corresponds to the new state, can be reached by infinitely large values. Unfortunately, mathematical infinity can not be handled by numerical algorithms. The desired behavior can be achieved by initializing the error covariance matrix P be achieved by a very large value instead of an infinite value. However, this can lead to numerical problems (for example, the larger the initial values are, the larger the problems are). The main cause is that some (or all) elements of the error covariance matrix P are simultaneously made up of small and very large components. Due to the limited numerical accuracy digital computer, the large components can wipe out the small components, which nevertheless can be important for accuracy. The initialization problem can be amplified by a model ambiguity. The model ambiguity can occur in run-to-run systems or similar systems, as described in more detail below.

5A shows an example to schematically illustrate the model ambiguity. Suppose that three products ( A . B and C ), all of which are handled by the same production tool. The observed quantity (eg, a polishing rate, a deposition rate, etc.) may indicate a synchronous movement over time, such as through the lines 500a . 500b . 500c for each of the products A . B . C is shown.

This behavior can be modeled by assuming a common tool state X _tool that passes through the line 500d to which product-specific preloads X _A , X _B and X _{C are} added. The individual states X _tool , X _A , X _B and X _C are not clearly defined by the data. As an example, the alternative state constellation that is in 5B illustrated as likely as the in 5A illustrated first constellation.

There are indeed infinite state constellations that can not be delimited by the data. There is an additional degree of freedom that is not resolved (determined) by the data and that can lead to estimation problems. The individual states X _tool , X _A , X _B and X _C are in contrast to the state combinations X _tool + X _A , X _tool + X _B and X _tool + X _C undefined. Because of this, the variance of the estimation error of the individual states is X _tool , X _A , X _B and X _{C are} infinite. This means that large components injected into the error covariance matrix P by initialization "with large values" will never disappear and therefore degrade the accuracy at each iteration step. According to various embodiments, these problems may be avoided, as described in more detail below, which may be a major improvement for the application of Kalman-based filtering to a pass-to-pass control system.

• • Das Vermeiden einer Modellmehrdeutigkeit durch das Definieren eines einzelnen Zustands für jede Kontextkombination, d. h. das Definieren eines eigenen Zustands für X_Werkzeug + X_A, X_Werkzeug + X_B bzw. X_Werkzeug + X_C: Dies kann eine einfache Lösung sein, wobei aber eine große Anzahl von Zuständen (z. B. Anzahl der Werkzeuge * Anzahl der Produkte * [...]) erzeugt werden kann. Ferner erfordert dies von null verschiedene Nichtdiagonalelemente in der Prozessrauschmatrix Q (die Speicherung und die Handhabung von n² Elementen). Im Fall von z. B. neuen Produkten müssen neue Zustände für alle Werkzeug-/Produktkombinationen sofort erzeugt werden; selbst wenn einige dieser Kombinationen niemals verwendet werden.
• • Das Eliminieren der Modellmehrdeutigkeit durch das Hinzufügen von Einschränkungen an die Zustände (wie z. B. X_A + X_B + X_C = 0): In diesem Fall wird eine verringerte Anzahl von Zuständen verwendet (Anzahl der Werkzeuge + Anzahl der Produkte + [...]), wobei die Prozessrauschmatrix Q diagonal sein kann (n anstelle von n² Elementen). Es können jedoch numerische Probleme auftreten und die Einschränkungen müssen regelmäßig aufgefrischt werden. Ferner können die Einschränkungen kompliziert zu entwerfen sein, insbesondere für komplexe Szenarios (z. B. für Mehrebenenzustände, Feinabstimmungs-Vorbelastungen, Gradienten, Polynome). Ferner sind im Fall von Dedikationen (z. B. eines Werkzeugs) einige Zustandskombinationen (z. B. Werkzeug/Produkt) undefiniert, wobei der Entwurf der Einschränkungen diese Tatsache berücksichtigen muss, die Einschränkungen dynamisch modifiziert werden müssen, falls die Werkzeug-Dedikation aufgegeben wird, und eine zusätzliche Logik benötigt wird, um die Zustände von den dedizierten Zuständen zu den nicht dedizierten Zuständen umzusetzen. Ferner sind die neuen Zustände während der Initialisierungsphase undefiniert, wobei diese Tatsache durch die Einschränkungen verborgen wird, wobei eine zusätzliche Logik benötigt wird, um diese Phase zu unterdrücken und/oder zu steuern.
• • Die Verwendung „mittelgroßer“ Anfangswerte: Dies kann eine komplizierte Herangehensweise sein. Der Anfangswert muss für die richtige Initialisierung des Filters groß genug sein und klein genug sein, um die numerischen Probleme unter Kontrolle zu halten. Die Anfangswerte müssen sehr sorgfältig eingestellt (z. B. „handgearbeitet“) werden, was für eine 24/7-Produktion und diverse Produktportfolios schwierig zu verwenden sein kann. Ferner ist in diesem Fall keine Unterscheidung zwischen nicht initialisierten und abgelaufenen Zuständen möglich. Ferner verschwinden die numerischen Probleme nicht vollständig. Es wird jedoch eine verringerte Anzahl von Zuständen (Anzahl der Werkzeuge + Anzahl der Produkte + [...]) verwendet, wobei die Prozessrauschmatrix Q diagonal sein kann (n anstelle von n² Elementen). Ferner können komplexe Szenarios (z. B. Mehrebenenzustände, Feinabstimmungs-Vorbelastungen, Gradienten, Polynome usw.) natürlich gehandhabt werden, wobei Zustandskombinationen definiert werden, wenn ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="E\; R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">E</figure-callout>}RR}^2<\text{Grenze}$
  
  gilt, wobei neue Zustandsebenen zur Laufzeit hinzugefügt werden können. Ferner können neue Zustände zur Laufzeit hinzugefügt und initialisiert werden und können Zustandskombinationen initialisiert werden, sobald ${\sigma }_{E\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}R}^2<\text{Grenze}$
  
  gilt. Ferner können Dedikationen (z. B. eines Werkzeugs) natürlich gehandhabt werden, wobei Zustandsdefinitionen definiert werden, falls ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="E\; R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">E</figure-callout>}RR}^2<\text{Grenze}$
  
  gilt, wobei, sobald nicht dedizierte Daten verfügbar sind, die dedizierten Zustände automatisch in die nicht dedizierten Zustände umgesetzt werden.

In general, there may be several strategies to overcome the problem outlined above, such as: B .:

• • Avoiding model ambiguity by defining a single state for each context combination, ie defining its own state for X _tool + X _A , X _tool + X _B, and X _tool + X _C, respectively: This may be a simple solution, but a large number of states (eg number of tools * number of products * [...]) can be generated. Furthermore, this requires non-zero non-diagonal elements in the process noise matrix Q (the storage and handling of n ² elements). In the case of z. For example, new products must immediately create new states for all tool / product combinations; even if some of these combinations are never used.
• • Eliminating the model ambiguity by adding constraints to the states (such as X _A + X _B + X _C = 0): In this case, a reduced number of states is used (number of tools + number of products + [...]), where the process noise matrix Q can be diagonal (n instead of n ² elements). However, there may be numerical problems and the restrictions must be refreshed regularly. Furthermore, the constraints may be complicated to design, especially for complex scenarios (eg, multi-level states, fine-tuning bias, gradients, polynomials). Further, in the case of dedications (eg, a tool), some state combinations (eg, tool / product) are undefined, and the design of constraints must take this fact into account, and the constraints must be dynamically modified if tool dedication is abandoned and additional logic is needed to translate the states from the dedicated states to the non-dedicated states. Furthermore, the new states are undefined during the initialization phase, this fact being hidden by the constraints, with additional logic needed to suppress and / or control that phase.
• • The use of "medium" initial values: this can be a complicated approach. The initial value must be large enough for proper initialization of the filter and small enough to keep the numerical problems under control. The initial values must be set very carefully (eg "handcrafted"), which can be difficult to use for a 24/7 production and various product portfolios. Furthermore, no distinction between uninitialized and expired states is possible in this case. Furthermore, the numerical problems do not completely disappear. However, there will be a reduced number of states (number of tools + number of products + [...] ), where the process noise matrix Q may be diagonal (n instead of n ² elements). Furthermore, complex scenarios (eg, multi-level states, fine-tuning preloads, gradients, polynomials, etc.) can be naturally handled, defining state combinations when ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2<\text{border}$
  
  applies, whereby new state levels can be added at runtime. Furthermore, new states can be added and initialized at runtime and state combinations can be initialized as soon as ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2<\text{border}$
  
  applies. Furthermore, dedications (eg, a tool) can be handled naturally, with state definitions defined if ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2<\text{border}$
  
  is true, as soon as dedicated data is not available, the dedicated states are automatically translated to the non-dedicated states.

According to various embodiments, the error covariance matrix P can be decomposed into a finite and an infinite part, which leads to all the advantages of the approach of the medium-sized initial values while avoiding the disadvantages. According to various embodiments, the error covariance matrix P may be decomposed as follows or in a similar manner: $$P_k=P_{R.k}+\kappa P_{l.k}.\kappa \to \infty$$

Gemäß verschiedenen Ausführungsformen kann sowohl zu dem endlichen als auch zu dem unendlichen Teil von P_k in dem Satz von Gleichungen (8) ein Prozessrauschen hinzugefügt werden. Dies führt zu einer modifizierten Gleichung der Gleichung (8*) im Folgenden. In diesem Fall können die folgenden Gleichungen entsprechend angepasst werden. $$\begin{array}{l}{\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+B_{k+1}\hfill \\ P_{R,k+1}=A_{k+1}{P}_{R,k}{A}_{k+1}^T+Q_{R,k+1}{Q}_{R,k+1}^T\hfill \\ P_{I,k+1}=A_{k+1}{P}_{I,k}{A}_{k+1}^T+Q_{I,k+1}{O}_{I,k+1}^T\hfill \end{array}$$

According to various embodiments, substituting equation (7) into equation (3) may result in a new set of process update equations (8) as follows: $$\begin{array}{l}{\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+B_{k+1}\hfill \\ P_{R.k+1}=A_{k+1}{P}_{R.k}{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T\hfill \\ P_{I.k+1}=A_{k+1}{P}_{l.k}{A}_{k+1}^T\hfill \end{array}$$

According to various embodiments, process noise may be added to both the finite and infinite parts of P _k in the set of equations (8). This leads to a modified equation of equation (8 *) below. In this case, the following equations can be adjusted accordingly. $$\begin{array}{l}{\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+B_{k+1}\hfill \\ P_{R.k+1}=A_{k+1}{P}_{R.k}{A}_{k+1}^T+Q_{R.k+1}{Q}_{R.k+1}^T\hfill \\ P_{I.k+1}=A_{k+1}{P}_{I.k}{A}_{k+1}^T+Q_{I.k+1}{O}_{I.k+1}^T\hfill \end{array}$$

$$F_{I,k+1}:=C_{k+1}{P}_{I,k}{C}_{k+1}^T$$

der Fall 1 und der Fall 2 wie folgt erhalten werden: Der Fall 1 kann durch |F_I,k+1| > 0 (in der Praxis durch |F_I,k+1| > F_NUM) definiert werden. In diesem Fall kann die Metrologieaktualisierung durch den folgenden Satz von Gleichungen (11) ausgedrückt werden: $$\begin{array}{c}{\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)\\ P_{R,k+1}=P_{R,k}-\frac{P_{R,k}{C}_{k+1}^T{C}_{k+1}{P}_{I,k}}{F_{I,k+1}}-\frac{P_{I,k}{C}_{k+1}^T{C}_{k+1}{P}_{R,k}}{F_{I,k+1}}+\frac{P_{I,k}{C}_{k+1}^T{C}_{k+1}{P}_{I,k}}{F_{I,k+1}^2/F_{R,k+1}}\\ P_{I,k}=P_{I,k}-\frac{P_{I,k}{C}_{k+1}^T{C}_{k+1}{P}_{I,k}}{F_{I,k+1}}\\ K_{k+1}=\frac{P_{I,k}{C}_{k+1}^T}{F_{I,k+1}}\end{array}$$

Der Fall 2 kann durch |F_I,k+1| = 0 (in der Praxis durch |F_I,k+1| ≤ L_NUM) definiert werden. In diesem Fall kann die Metrologieaktualisierung durch den folgenden Satz von Gleichungen (12) ausgedrückt werden: $$\begin{array}{c}{\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)\\ P_{R,k+1}=P_{R,k}-\frac{P_{R,k}{C}_{k+1}^T{C}_{k+1}{P}_{R,k}}{F_{R,k+1}}\\ P_{I,k+1}=P_{I,k}\\ K_{k+1}=\frac{P_{R,k}{C}_{k+1}^T}{F_{R,k+1}}\end{array}$$

If the error covariance matrix P z. For example, according to equation (7), the metrology update (eg, according to equation (4)) may be decomposed into two cases. According to various embodiments, based on the following equations: $$F_{R.k+1}:=C_{k+1}{P}_{R.k}{C}_{k+1}^T+R_{k+1}^2$$

$$F_{I.k+1}:=C_{k+1}{P}_{I.k}{C}_{k+1}^T$$

the case 1 and the case 2 can be obtained as follows: The case 1 can by | F _{I, k + 1} | > 0 (in practice by | F _{I, k + 1} |> F _NUM ). In this case, the metrology update can be expressed by the following set of equations (11): $$\begin{array}{c}{\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)\\ P_{R.k+1}=P_{R.k}-\frac{P_{R.k}{C}_{k+1}^T{C}_{k+1}{P}_{I.k}}{F_{I.k+1}}-\frac{P_{I.k}{C}_{k+1}^T{C}_{k+1}{P}_{R.k}}{F_{I.k+1}}+\frac{P_{I.k}{C}_{k+1}^T{C}_{k+1}{P}_{I.k}}{F_{I.k+1}^2/F_{R.k+1}}\\ P_{I.k}=P_{I.k}-\frac{P_{I.k}{C}_{k+1}^T{C}_{k+1}{P}_{I.k}}{F_{I.k+1}}\\ K_{k+1}=\frac{P_{I.k}{C}_{k+1}^T}{F_{I.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}}\end{array}$$

The case 2 can by | F _{I, k + 1} | = 0 (in practice defined by | F _{I, k + 1} | ≤ L _NUM ). In this case, the metrology update can be expressed by the following set of equations (12): $$\begin{array}{c}{\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)\\ P_{R.k+1}=P_{R.k}-\frac{P_{R.k}{C}_{k+1}^T{C}_{k+1}{P}_{R.k}}{F_{R.k+1}}\\ P_{I.k+1}=P_{I.k}\\ K_{k+1}=\frac{P_{R.k}{C}_{k+1}^T}{F_{R.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}}\end{array}$$

für die Vorhersage irgendeiner lineare Zustandskombination von Interesse ${\widehat{\alpha }}_{k+1|k}^2=\overline{C}{\widehat{x}}_k$

ist z. B. durch Folgendes gegeben:The numerical limit, L _NUM , can be z. B. to a very small number, z. To a value of 10 ^-10 . The error variance ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="e\; R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">e\; </figure-callout>}RR}^2$

for the prediction of any linear state combination of interest ${\widehat{\alpha }}_{k+1|\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2=\overline{C}{\widehat{x}}_k$

is z. For example, given by the following:

$$\begin{array}{c}{\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2=\{\begin{array}{cc}\overline{C}\left[P_{R.k}+Q_{k+1}{Q}_{k+1}^T\right]{\overline{C}}^T& \text{if}{F}_I\le {\text{L}}_{\text{NUM}}\\ \infty & \text{if}{F}_I{\text{> L}}_{\text{NUM}}\end{array}.\text{With}\\ F_I=\overline{C}{P}_{I.k}{\overline{C}}^T\end{array}$$

für die Vorhersage des nächsten Methodologiewertes ŷ_k+1|k = C_k+1x̂_k ist z. B. durch Folgendes gegeben: $$\begin{array}{c}{\sigma }_{ERR}^2=\{\begin{array}{cc}{C}_{k+1}{P}_{R,k}{C}_{\text{k}+1}^{\text{T}}+R_{k+1}^2& \text{falls }{F}_I\le {\text{L}}_{\text{NUM}}\\ \infty & \text{falls }{F}_I{\text{>L}}_{\text{NUM}}\end{array},\text{mit}\\ F_I=C_{k+1}{P}_{I,k}{C}_{k+1}^{\text{T}}\end{array}$$

The error variance ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="e\; R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">e\; </figure-callout>}RR}^2$

for the prediction of the next methodological value ŷ _{k + 1 | k} = C _{k + 1} x _k is z. For example, given by the following: $$\begin{array}{c}{\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2=\{\begin{array}{cc}{C}_{k+1}{P}_{R.k}{C}_{\text{k}+1}^{\text{T}}+R_{k+1}^2& \text{if}{F}_I\le {\text{L}}_{\text{NUM}}\\ \infty & \text{if}{F}_I{\text{> L}}_{\text{NUM}}\end{array}.\text{With}\\ F_I=C_{k+1}{P}_{I.k}{C}_{k+1}^{\text{T}}\end{array}$$

• • eine exakte Initialisierung mit unendlich „∞“ ist möglich,
• • der Anfangswert kann beliebig gewählt werden, weil das Ergebnis unabhängig von dieser Wahl ist,
• • die Rechenkomplexität der Prozessaktualisierung ohne Zustandsübergang (d. h., A_k+1 =I_n) ist von n,
• • die Rechenkomplexität der Metrologieaktualisierung ist von n² (falls sie richtig implementiert ist, wie z. B. im Folgenden beschrieben wird),
• • nicht initialisierte und abgelaufene Zustände können voneinander abgegrenzt werden,
• • aufgrund zu „kleiner“ und zu „großer“ Komponenten in der Fehlerkovarianzmatrix P treten keine numerischen Probleme auf,
• • es wird eine verringerte Anzahl von Zuständen verwendet (Anzahl der Werkzeuge + Anzahl der Produkte + [...]),
• • die Prozessrauschmatrix Q kann diagonal sein (n anstelle von n² Elementen),
• • komplexe Szenarios (z. B. Mehrebenenzustände, Feinabstimmungs-Vorbelastungen, Gradienten, Polynome) werden natürlich gehandhabt:
  • ■ es werden Zustandsdefinitionen definiert, falls ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="E\; R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">E</figure-callout>}RR}^2<\infty$
    
    gilt,
  • ■ neue Zustandsebenen können zur Laufzeit hinzugefügt werden, neue Zustände können zur Laufzeit hinzugefügt und initialisiert werden,
  • ■ Zustandskombinationen werden initialisiert, sobald ${\sigma }_{E\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}R}^2<\infty$
    
    gilt,
• • Dedikationen (z. B. eines Werkzeugs) werden natürlich behandelt:
  • ■ Zustandsdefinitionen werden definiert, falls ${\sigma }_{\mathrm{<figure-callout\; id="2"\; label="E\; R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">E</figure-callout>}RR}^2<\infty$
    
    gilt,
  • ■ sobald nicht dedizierte Daten verfügbar sind, werden die dedizierten Zustände automatisch in nicht dedizierte Zustände umgesetzt.

This approach can accomplish one or more of the following:

• • an exact initialization with infinite "∞" is possible
• • the initial value can be chosen arbitrarily, because the result is independent of this choice,
• • The computational complexity of the process update without state transition (ie, A _{k + 1} = I _n ) is of n .
• The computational complexity of the metrology update is of n ² (if properly implemented, eg as described below),
• • uninitialized and expired states can be separated from each other,
• Due to "small" and "big" components in the error covariance matrix P there are no numerical problems
• • a reduced number of states is used (number of tools + number of products + [...]),
• • the process noise matrix Q can be diagonal ( n instead of n ² elements),
• • complex scenarios (eg multi-level states, fine-tuning preloads, gradients, polynomials) are handled naturally:
  • ■ state definitions are defined if ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2<\infty$
    
    applies,
  • ■ new state levels can be added at runtime, new states added and initialized at runtime,
  • ■ State combinations are initialized as soon as ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2<\infty$
    
    applies,
• • Dedications (eg of a tool) are handled naturally:
  • ■ State definitions are defined if ${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2<\infty$
    
    applies,
  • ■ Unless dedicated data is available, the dedicated states are automatically translated to non-dedicated states.

However, a specification of a numerical boundary L _NUM may be complicated, especially if the range of the observation vector _elements C _k (i) is large, which will be described in more detail below.

According to various embodiments, an initialization problem of the run-to-run Kalman filter may be achieved by decomposing the error covariance matrix P into a finite part P _R and an infinite part P _I prevented as described above. Further, the problems due to model ambiguity may be due to the error covariance matrix P into a finite part P _R and an infinite part P _I prevented as described above.

According to various embodiments, the modification of the Kalman filter described above may be part of a universal pass-to-run / monitor system used in semiconductor (or other) industries. In the case where a Kalman model is used and in which the error covariance matrix is decomposed into a finite and an infinite part, the operation of the data merging circuit can 108 be improved. The iterative equations described above can lead to a characteristic sequence of the state estimation x _k and the error covariance matrix P _{R, k} , P _{I, k} k = 0, 1, ... lead.

According to various embodiments, the data fusion circuit 108 be configured to determine the accuracy information based on an estimation error, wherein the estimation error similar to the above-described division of the error covariance matrix P can be decomposed into a finite part and an infinite part. The estimation error may be via an error covariance matrix, e.g. Based on the error covariance matrix P or based on an inverted error covariance matrix P ^-1 . In this case, the error covariance matrix becomes P or the inverted error covariance matrix P ^-1 decomposes.

As a result, the data merging circuit 108 Further, it may be configured to reconstruct the prior state of state information X based on a first set of equations for restoring the prior state (eg, the set of equations (11)) to a first case and to a second set of equations for restoring the previous one State (eg, the set of equations (12)) in a second case. The first and second cases are determined by a distinguishing criterion, e.g. Based on the equation (10), where | F _{I, k + 1} | > L _NUM defines the first case and | F _{I, k + 1} | ≤ L _NUM defines the second case. However, there may be other values and similar definitions for distinguishing the first case from the second case.

According to various embodiments, the data fusion circuit 108 be configured to calculate the estimated measurement data and a corresponding prediction error and to check whether the prediction error contains infinite parts.

According to various embodiments, the earlier state of the state information for one measurement vector may be recovered over a plurality of single prior art scalar recoveries, wherein for each individual scalar restoration of the previous state, e.g. Based on a numerical reference value, L _NUM , as described above, it is checked whether the discrimination criterion is zero.

According to various embodiments, for a infinity filter, a fast and stable quadratic algorithm for restoring the prior state is provided. The state estimation may be provided by a control algorithm included in the data fusion circuit 108 the process control circuit 100 is implemented. The control algorithm may be provided as described above or provided based on similar equations describing the same general approach. According to various embodiments, a Kalman filter or a Kalman based filter (ie, a modification or extension of the standard Kalman filter) may be advantageously used to provide the state estimate. In the following, various aspects will be described which are applicable to control any described process, e.g. G., By the equations (1), (1 *), (2), (2 *), or in a similar manner as described above.

$$P_{R,k+1}=A_{k+1}{P}_{R,k}{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T,$$

$$P_{I,k+1}=A_{k+1}{P}_{I,k}{A}_{k+1}^T.$$

In more detail, the state information of a process, e.g. Modulated by equations (1), (2) can be estimated by two sets of equations that are performed completely independently of one another, see e.g. For example, the set of equations (3), (4) has been described above. After each process or maintenance step, a process update is performed according to the first set of equations (8) using equations (8a) to (8c): $${\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+{\mathrm{<figure-callout\; id="1"\; label="B\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_{k+}.$$

$$P_{R.k+1}=A_{k+1}{P}_{R.k}{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T.$$

$$P_{I.k+1}=A_{k+1}{P}_{I.k}{A}_{k+1}^T,$$

$$F_{I,k+1}:=C_{k+1}{P}_{I,k}{C}_{k+1}^T.$$

Im Fall 1, der z. B. durch |F_I,k+1| > 0 oder |F_I,k+1| > F_NUM definiert ist, kann die Metrologieaktualisierung basierend auf dem folgenden Satz von Gleichungen (11), der die Gleichungen (11a) bis (11d) enthält, ausgeführt werden: $${\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right),$$

$$P_{R,k+1}=P_{R,k}-\frac{P_{R,k}{C}_{k+1}^T{C}_{k+1}{P}_{I,k}}{F_{I,k+1}}-\frac{P_{I,k}{C}_{k+1}^T{C}_{k+1}{P}_{R,k}}{F_{I,k+1}}+\frac{P_{I,k}{C}_{k+1}^T{C}_{k+1}{P}_{I,k}}{F_{I,k+1}^2/F_{R,k+1}},$$

$$P_{I,k}=P_{I,k}-\frac{P_{I,k}{C}_{k+1}^T{C}_{k+1}{P}_{I,k}}{F_{I,k+1}},$$

$$K_{k+1}=\frac{P_{I,k}{C}_{k+1}^T}{F_{I,k+1}}.$$

Im Fall 2, der z. B. durch |F_I,k+1| = 0 oder |F_I,k+1| ≤ L_NUM definiert ist, kann die Metrologieaktualisierung basierend auf dem folgenden Satz von Gleichungen (12), der die Gleichungen (12a) bis (12d) enthält, ausgeführt werden: $${\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right),$$

$$P_{R,k+1}=P_{R,k}-\frac{P_{R,k}{C}_{k+1}^T{C}_{k+1}{P}_{R,k}}{F_{R,k+1}},$$

$$P_{I,k+1}=P_{I,k},$$

$$K_{k+1}=\frac{P_{R,k}{C}_{k+1}^T}{F_{R,k+1}}.$$

Furthermore, metrology data can be updated by the metrology update, e.g. Also referred to as restoration of the previous state, to which state information is added, as described above with the following definition: $$F_{R.k+1}:=C_{k+1}{P}_{R.k}{C}_{k+1}^T+R_{k+1}^2.$$

$$F_{I.k+1}:=C_{k+1}{P}_{I.k}{C}_{k+1}^T,$$

In the case 1 , the z. By | F _{I, k + 1} | > 0 or | F _{I, k + 1} | > F _NUM , the metrology update may be performed based on the following set of equations (11) containing equations (11a) through (11d): $${\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right).$$

$$P_{R.k+1}=P_{R.k}-\frac{P_{R.k}{C}_{k+1}^T{C}_{k+1}{P}_{I.k}}{F_{I.k+1}}-\frac{P_{I.k}{C}_{k+1}^T{C}_{k+1}{P}_{R.k}}{F_{I.k+1}}+\frac{P_{I.k}{C}_{k+1}^T{C}_{k+1}{P}_{I.k}}{F_{I.k+1}^2/F_{R.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}}.$$

$$P_{I.k}=P_{I.k}-\frac{P_{I.k}{C}_{k+1}^T{C}_{k+1}{P}_{I.k}}{F_{I.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}}.$$

$$K_{k+1}=\frac{P_{I.k}{C}_{k+1}^T}{F_{I.k+1}},$$

In the case 2 , the z. By | F _{I, k + 1} | = 0 or | F _{I, k + 1} | ≤ L _NUM , the metrology update may be performed based on the following set of equations (12) containing equations (12a) through (12d): $${\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right).$$

$$P_{R.k+1}=P_{R.k}-\frac{P_{R.k}{C}_{k+1}^T{C}_{k+1}{P}_{R.k}}{F_{R.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}}.$$

$$P_{I.k+1}=P_{I.k}.$$

$$K_{k+1}=\frac{P_{R.k}{C}_{k+1}^T}{F_{R.k+1}},$$

If the metrology data consists of more than one measurement value, the update can be performed separately with equations (9) - (12d) for each value, as described above. The sequence of equations (8a) - (8c) and (9) - (12d) accurately reflects the accumulation and degradation of the process information (e.g., the reduction and increase in estimation error) as long as the process model parameters Q, R, B, D, C, A fit the process in real life.

According to various embodiments, the estimation algorithm (eg, according to equations (8a) - (12d)) may have advantages over conventional Kalman filtering (especially if it is applied to a pass-to-pass control). Distinguishing the case 1 and the case 2 however, in the metrology update of each other may be an important aspect of the algorithm, as described in more detail below.

As an example shows 6A the distinction parameter F _I for a model with 230 states over 1000 iterations, wherein the distinguishing parameter is plotted on the y-axis 600y and the iterations are plotted on the x-axis 600x. The elements of the observation vector, C , are only 1 and 0, see equation (10). As in 6A is illustrated, are the case 1 with (| F _I | ≈ 1.0) and the case 2 with (| F _I | ≈ 10 ^-15 ) well separated from each other in this example, with a discrimination threshold surely being e.g. B. can be set to L _NUM = 10 ^-10 (see the distinction line 600d ), where the basic numerical noise floor 600n is about 10 ^-16 .

If the elements of the observation vector, C , over a slightly wider area, eg. B. (0.01, 1.0, 1000) are distributed, the distinguishing parameter F _{I is} distributed more complex, such. In 6B is illustrated. The first case is now characterized by the values of the distinguishing parameter F _I , e.g. B. from 10 ⁶ to 10 ^-4 range. The second case corresponds to the values of the discrimination parameter of about 10 ^-10 , e.g. With | F _I | ≈ 10 ^-10 , (for example, due to the basic numerical noise floor 600n of about 10 ^-10 ), leaving little room for a clear distinction between the two cases and therefore between the application of equations (11a) - (11d) or equations (12a) - (12d). In the 6A illustrated safe discrimination _limit L _NUM = 10 ^-10 would be inappropriate in this case, which in turn means that L _NUM would have to be set individually for each use case. In accordance with various embodiments, a state estimation algorithm is provided below to avoid the need for individually setting the discrimination limit L _NUM . This algorithm (also referred to as the square root algorithm) may be used in the process control circuit 100 or the data fusion circuit 108 an automatic control system, the z. B. operated on a 24/7 basis can be used, as described above.

$$P_I=S_I^T{S}_I,$$

In diesem Fall werden anstelle der Fehlerkovarianzmatrizen P_R, P_I die Quadratwurzelmatrizen S_R, S_I direkt ausgebreitet. In diesem Fall werden die Quadratwurzelmatrizen S_R, S_I z. B. außerdem als Cholesky-Faktoren bezeichnet.According to various embodiments, a square root algorithm is provided based on equations (8a) - (12d), as described in more detail below. According to various embodiments, the covariance matrices P _R and P _{I are} replaced by the factorizations. The factorization can z. B. be defined as follows or in a similar manner: $$P_R=S_R^T{S}_R.$$

$$P_I=S_I^T{S}_I.$$

In this case, instead of the error covariance matrices P _R , P _I, the square root matrices S _R , S _{I are} directly propagated. In this case, the square root matrices S _R , S _I z. B. also referred to as Cholesky factors.

In diesem Fall ist U z. B. ein verallgemeinerter Cholesky-Faktor und ist D eine Diagonalmatrix.Although the following embodiment will be described based on a specific example of factorization, see Equations (15), (16), other factorizations may be used in the same way as e.g. B .: $$P=U^{T}DU$$

In this case, U z. For example, a generalized Cholesky factor and D is a diagonal matrix.

Furthermore, the following factorizations can be used in a similar way: $$P=S{S}^T\text{or}P=UD{U}^T$$

According to various embodiments, the strategy of radying may be applied to Kalman filtering. The condition number of the square roots S _R , S _I (the Cholesky factors) is better than the condition number of the original matrices P _R , P _I. The numerical challenging part may be the restoration of the previous state (the metrology update). For standard Kalman filters, the most accurate (and at the same time the most efficient) algorithms can be the Carlson algorithm and the Bierman algorithm. Both the Carlson algorithm and the Bierman algorithm are based on the same principles and are almost equivalent from a numerical standpoint. According to various embodiments, the Carlson / Bierman algorithm is implemented in a modified, e.g. B. generalized, way for the restoration of the previous state z. Based on equations (11a) - (12d). The implementation of the modified Carlson algorithm is shown in more detail below by way of example.

According to various embodiments, the rubbing of equations (8a) - (12d) may encounter one or more of the following aspects: process update (8b), (8c) ■ Kalman upgrade ■ Complexity ~ n ³ metrology update (12b) - (12d) ■ Kalman recovery of the previous state ■ The Carlson algorithm is applicable The Carlson algorithm does not work for singular matrices metrology update (11b) ■ coupled restore of the previous state of rank 1 and update of rank 2 ■ sequential execution of the previous state restores and Rank 1 updates will cause problems due to numeric mismatch metrology update (11c), (11d) ■ "Total restoration of the previous state" → reduces the rank of P _I ■ The Carlson algorithm is not designed for a total restoration of the previous state ■ P _I is singular after the first restoration of the previous state ■ The Carlson algorithm does not work for singular matrices

geschrieben werden können. Die Matrizen S und S sind jeweils obere Dreiecksmatrizen. Der Algorithmus kann nicht angewendet werden, wenn R = 0 gilt (z. B. im Fall einer „totalen Wiederherstellung des früheren Zustands“), oder wenn die Eingangsmatrix S singulär ist. Es kann jedoch beides für das Radizieren der Gleichung (11c und 12b) zur Wiederherstellung des früheren Zustands notwendig sein. Deshalb wird gemäß verschiedenen Ausführungsformen der Carlson-Algorithmus modifiziert, wie im Folgenden ausführlicher beschrieben wird.According to various embodiments, the Carlson algorithm may be used for the steps of restoring the previous state, e.g. For example, as follows: $${\overline{S}}^T\overline{S}=S^T\left(I-\frac{a{a}^T}{a^{T}a+R^2}\right)S$$

can be written. The matrices S and S are each upper triangular matrices. The algorithm can not be applied if R = 0 (eg in the case of a "total restoration of the previous state"), or if the input matrix S is singular. However, both may be necessary for the erasing of the equation (11c and 12b) to restore the former state. Therefore, according to various embodiments, the Carlson algorithm is modified, as described in more detail below.

The generalized restoration of the previous state based on the modified Carlson may be e.g. With algorithm (1), which is referred to as the Generalized Carlson Recovery of Earlier State (GCDD), as follows:

ALGORITHM 1:

• • es können sowohl singulärer Matrizen als auch totale Wiederherstellungen des früheren Zustands (R = 0) behandelt werden;
• • jede totale Wiederherstellung des früheren Zustands erzeugt eine neue Zeile, die exakt null ist;
• • ein Rangdefizit wird in der Matrixstruktur codiert:
  • ■ es vergrößert die numerische Genauigkeit,
  • ■ Zeilen, die exakt null sind, müssen nicht gespeichert werden,
  • ■ P_I weist typischerweise nur einige von null verschiedene Zeilen auf, wobei deshalb
  • ■ der Speicherraum für P = P_R + P_I um etwa 50 % verringert ist;
• • alle von null verschiedenen Zeilen weisen von null verschiedene Diagonalelemente auf;
• • Zeilen, die exakt null sind, werden anders als von null verschiedene Zeilen verarbeitet; und
• • Zeilen, die exakt null sind, werden während der weiteren Verarbeitung bewahrt.

The generalized recovery of the previous state, based on the algorithm provided above or implemented in a similar manner, may have one or more of the following properties:

• • both singular matrices and total restores of the previous state (R = 0) can be treated;
• • each total restore of the previous state creates a new line that is exactly zero;
• • a ranking deficit is coded in the matrix structure:
  • ■ it increases the numerical accuracy,
  • ■ Lines that are exactly zero do not need to be saved
  • ■ P _I typically has only a few nonzero lines, therefore
  • ■ the memory space for P = P _R + P _I is reduced by about 50%;
• • all nonzero lines have non-zero diagonal elements;
• • Lines that are exactly zero are processed differently than non-zero lines; and
• • Lines that are exactly zero are preserved during further processing.

6C illustrates the structure 602 the matrix S according to various embodiments. As illustrated, the rank deficit is structural in the matrix S coded. As described above, each full recovery of the previous state creates a new line 602z that are exactly zero, with the lines 602z , which are exactly zero, of the matrix S are preserved during further processing. All nonzero lines 602r the matrix S have non-zero diagonal elements 602d on.

Compared to the respect 6B provided example, which contains the unmodified algorithm, the distribution of the values of the distinguishing parameter F _I , which is generated by the modified algorithm, in 6D illustrated. As illustrated, this is the case 1 still characterized by the values of the distinguishing parameter F _I ranging from 10 ⁶ to 10 ^-4 . The numerical basic noise 602n however, is reduced by several orders of magnitude to about 10 ^-33 , which is a safe discrimination threshold above the basic numerical noise floor 602n , z. Eg L _NUM = 10 ^-25 (see the distinguishing line 602d ).

In addition, the algorithm described above ( 1 ) the ability to add exact linear constraints to a Kalman-based process model. The linear constraints may be expressed as one or more measurement equations (eg, similar to equation (2)) with R = 0 and therefore correspond to total restores of the previous state. Conventional filters may not be able to handle exact linear constraints and therefore require an approximation of the exact constraint of "small" measurement equations. R that are greater than zero. Because in a conventional filter, the exact constraint noise parameter R can be emulated with a "small" number, such a conventional filter can lead to serious numerical problems.

Um die Gleichung (18) in die Gleichung (17) zu transformieren, kann die Matrix S aus v durch das Lösen der folgenden Gleichung für a ausgeklammert werden: $$v=S^{T}a$$

Dies ist in dem Fall möglich, in dem der Vektor v zum Bild der Matrix S^T gehört. Diese Bedingung ist z. B. erfüllt, wenn S und S positiv semidefinit sind. Das Einsetzen der Gleichung (19) in die Gleichung (18) führt z. B. zu einer Gleichung (20) zur Wiederherstellung des früheren Zustands wie folgt: $${\overline{S}}^T\overline{S}=S^{T}S-v{v}^T=S^T\left(I-a{a}^T\right)S=S^T\left(I-\frac{a{a}^T}{\underset{=:R^2}{\underbrace{1-a^{T}a}}+a^{T}a}\right)S$$

According to various embodiments, the algorithm (1) described above may be e.g. B. extended to any restores of the previous state, based on the following equation: $${\overline{S}}^T\overline{S}=S^{T}S-v{v}^T,$$

To transform equation (18) into equation (17), the matrix may be S are excluded from v by solving the following equation for a: $$v=S^{T}a$$

This is possible in the case where the vector v belongs to the image of the matrix S ^T. This condition is z. B. fulfilled if S and S are positive semidefinite. Substituting equation (19) into equation (18) results in e.g. To an equation (20) for restoring the previous state as follows: $${\overline{S}}^T\overline{S}=S^{T}S-v{v}^T=S^T\left(I-a{a}^T\right)S=S^T\left(I-\frac{a{a}^T}{\underset{=:{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2}{\underset{\}}{1-a^{T}a}}+a^{T}a}\right)S$$

This gives rise to the following modified state Carlson reconstruction algorithm as follows, according to various embodiments:

ALGORITHM 2

1. 1. Input:
   • S - upper (n, n) triangular matrix
   • v (n, 1) vector
   • tol - scalar
2. 2. Solve: v = S ^T a for a; z. For example, a = (S ^T ) ⁺ v
3. 3. Calculate $R=\sqrt{1-a^{T}a}$
   
4. 4. The result matrix S calculate with the algorithm (1):
   • S = GCDD (S, a, R, tol)

und mit $$\begin{array}{ccc}{\overline{P}}_R:=P_{R,k+1;}& P_R:=P_{R,k};& P_I:=P_{I,k};\\ F_R:=F_{R,k+1};& F_I:=F_{I,k+1};& C_{k+1}:=C\end{array}$$

In accordance with various embodiments, two independent highly stable algorithms for rendering e.g. The equations (11b) to (11d), see the algorithm (3) and the algorithm (5) below. For the sake of brevity, equation (11b) may be e.g. B. be rewritten as follows: $$\begin{array}{c}{\overline{P}}_R=P_R-\frac{P_R{C}^{T}C{P}_I}{F_I}-\frac{P_I{C}^{T}C{P}_R}{F_I}+\frac{P_I{C}^{T}C{P}_I}{F_I^2}{F}_R.\text{With}\\ F_R:=P_{R}C{P}_R{C}^T+{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2\end{array}$$

and with $$\begin{array}{ccc}{\overline{P}}_R:=P_{R.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1;}& P_R:=P_{R.k};& P_I:=P_{I.k};\\ F_R:=F_{R.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1};& F_I:=F_{I.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1};& C_{k+1}:=C\end{array}$$

$${\overline{S}}^T{\overline{S}}_R=\left[{\left(I-\frac{C^{T}C{P}_I}{F_I}\right)}^T{S}_R^T\frac{P_I{C}^{T}R}{F_I}\right]\cdot \left[\begin{array}{c}{S}_R{\left(I-\frac{C^{T}C{P}_I}{F_I}\right)}^T\\ \frac{C{P}_{I}R}{F_I}\end{array}\right]=M^{T}M$$

The basic idea of the algorithm (3) is that the equation (21) z. B. to rearrange as follows: $${\overline{P}}_R={\left(I-\frac{C^{T}C{P}_I}{F_I}\right)}^T\underset{S_R^T{S}_R}{\underset{\}}{P_R}}\left(I-\frac{C^{T}C{P}_I}{F_I}\right)+\frac{P_I{C}^{T}C{P}_I}{F_I^2}{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2$$

$${\overline{S}}^T{\overline{S}}_R=\left[{\left(I-\frac{C^{T}C{P}_I}{F_I}\right)}^T{S}_R^T\frac{P_I{C}^{T}R}{F_I}\right]\cdot \left[\begin{array}{c}{S}_R{\left(I-\frac{C^{T}C{P}_I}{F_I}\right)}^T\\ \frac{C{P}_{I}R}{F_I}\end{array}\right]=M^{T}M$$

According to various embodiments, the matrix may be factorized using a QR decomposition as follows: $$M=T\left[\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">M</figure-callout>}}_1\\ 0\end{array}\right];T^{T}T=I$$

Substituting equation (25) into equation (24) shows that the result matrix S _{R is given} by the upper triangular matrix M _I as follows: $${\overline{S}}_R^T{\overline{S}}_R=\left[\begin{array}{cc}{M}_1^T& 0\end{array}\right]\underset{I}{\underset{\}}{T^{T}T}}\left[\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">M</figure-callout>}}_1\\ 0\end{array}\right]=M_1^{\mathrm{<figure-callout\; id="1"\; label="T\; M"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">T\; </figure-callout>}}{M}_{}{}_1+0$$

According to various embodiments, z. For example, based on the algorithm (3), the square root of equations (11b) to (11d) are formed:

ALGORITHM 3:

• 1. Input:
  • S _{R, k} - upper (n, n) triangular matrix
  • S _{I, k} - upper (n, n) triangular matrix
  • C _{k + 1} - (1, n) observation vector
  • R - scalar
• 2. Calculate: $m_R:=S_{R.k}{C}_{k+1}^T;m_I:=S_{I.k}{C}_{k+1}^T;u_I:=S_{I.k}^T{m}_I;F_I:=m_I^T{m}_I$
  
• 3. Build the matrix: $M:=\left[\begin{array}{c}{S}_{R.k}-\left(m_R{u}_I^T\right)/F_I\\ u_I^T\left(R/F_I\right)\end{array}\right]$
  
• 4. Use orthogonal row transforms (eg, Givens rotations) to eliminate the last row of matrix M and restore the upper triangular shape of the other rows. Ensure that all lines that are exactly zero are preserved, and that all nonzero lines have a non-zero diagonal element. $$\overline{M}=\left[\begin{array}{c}{S}_{R.k+1}\\ 0\end{array}\right]=TM$$
  
• 5. The result matrix S _{R, k + 1} is defined by lines 1, 2,..., N of M given.
• 6. Calculate the result matrix S _{I, k + 1} and K _{k + 1} with the algorithm (1): $\lfloor S_{I.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}.K_{k+1}\rfloor =\text{GCDD}\left(S=S_{I.k}.a=m_I.R=0\text{tol = 0}\right)$
  

Due to the orthogonal row transformation in the point 4 of the algorithm (3), the computational complexity of the algorithm (3) is proportional to n ³ . This can not be a problem because equations (11a) to (11d) are generally performed very rarely (eg, at most n times).

According to various embodiments, based on the algorithm (5), the square root of the equations (11b) to (11d) may be z. B. be formed as follows. Algorithm (5) separates equation (11b) into three sequential updates of rank 1 and a restoration of the former state of rank 1 , This can be based on the following equation (27): $$\begin{array}{cc}C{\overline{P}}_R{C}^T& =C\left[P_R-\frac{P_R{C}^{T}C{P}_I}{F_I}-\frac{P_I{C}^{T}C{P}_R}{F_I}+\frac{P_I{C}^{T}C{P}_I}{F_I^2}\underset{{\mathrm{<figure-callout\; id="0"\; label="F\; R"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">F\; </figure-callout>}}_R+R^2}{\underset{\}}{F_R}}\right]C^T\\ & =\underset{=0}{\underset{\}}{C\left[P_R-\frac{P_R{C}^{T}C{P}_I}{F_I}-\frac{P_I{C}^{T}C{P}_R}{F_I}+\frac{P_I{C}^{T}C{P}_I}{F_I^2}{F}_{R0}\right]C^T}}\\ & +\underset{={\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2}{\underset{\}}{C\left[\frac{P_I{C}^{T}C{P}_I}{F_I^2}{R}^2\right]C^T}}\\ & ={\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2\\ & \text{With}{\mathrm{<figure-callout\; id="0"\; label="F\; R"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">F\; </figure-callout>}}_R=C{P}_R{C}^T\end{array}$$

• 1. Aktualisierung des Rangs 1 mit dem Vektor $\left[P_I{C}^T\right]\frac{\sqrt{F_{R0}}}{F_I}$
  
• 2. Aktualisierung des Rangs 1 mit dem „positiven“ Eigenvektor von $$\frac{P_R{C}^{T}C{P}_I}{F_I}+\frac{P_I{C}^{T}C{P}_R}{F_I}$$
  
• 3. Wiederherstellung des früheren Zustands des Rangs 1 mit dem „negativen“ Eigenvektor von $\frac{P_R{C}^{T}C{P}_I}{F_I}+\frac{P_I{C}^{T}C{P}_R}{F_I}$
  
• 4. Aktualisierung des Rangs 1 mit dem Vektor $\left[P_I{C}^T\right]\frac{R}{F_I}$
  

The left side of equation (21) can be calculated by the procedure listed below. Equation (27) shows that the step 3 in this process a total restoration of the previous state is:

• 1. Update the rank 1 with the vector $\left[P_I{C}^T\right]\frac{\sqrt{F_{R0}}}{F_I}$
  
• 2. Update the rank 1 with the "positive" eigenvector of $$\frac{P_R{C}^{T}C{P}_I}{F_I}+\frac{P_I{C}^{T}C{P}_R}{F_I}$$
  
• 3. Restore the former state of rank 1 with the "negative" eigenvector of $\frac{P_R{C}^{T}C{P}_I}{F_I}+\frac{P_I{C}^{T}C{P}_R}{F_I}$
  
• 4. Update the rank 1 with the vector $\left[P_I{C}^T\right]\frac{R}{F_I}$
  

According to various embodiments, the total restoration of the previous state in step 3 in terms of by the step 1 and the step 2 of the sequence generated result matrix. This can be z. B. ensure numerical stability.

For the steps 1 . 2 and 4 of the expiration can be an algorithm of updating the rank 1 be used. This algorithm of updating the rank 1 can be modified slightly to preserve the lines that are exactly zero in S _R , such as. B. in the algorithm (4) is shown. Furthermore, the algorithm (1) for the step 3 of the process.

According to various embodiments, updating the rank 1 . S ^T S = S ^T S + aa ^T , which is called R1UD, can be carried out as follows:

ALGORITHM 4

• 1. Input:
  • S - upper (n, n) triangular matrix
  • a (n, 1) vector
• 2. Build the matrix: $M:=\left[\begin{array}{c}S\\ a^T\end{array}\right]$
  
• 3. Use the orthogonal row transforms (eg Givens rotations) to eliminate the last row of M's. Ensure that all lines that are exactly zero are preserved and that all non-zero lines have a non-zero diagonal element. $$M:=\left[\begin{array}{c}\overline{S}\\ 0\end{array}\right]=TM$$
  
• 4. The result matrix S is through the lines 1 . 2 , ..., n of M given.

According to various embodiments, the algorithm (5) shown below may be used exclusively for ranked updates 1 and restoring the former state of rank 1 be used. The computational complexity in this case is proportional to n ² .

ALGORITHM 5:

• 1. Input:
  • S _{R, k} - upper (n, n) triangular matrix
  • S _{I, k} - upper (n, n) triangular matrix
  • C _{k + 1} - (1, n) observation vector
  • R - scalar
  2. Calculate: $m_R:=S_{R.k}{C}_{k+1}^T;m_I:=S_{I.k}{C}_{k+1}^T;{\text{F}}_{R0}:=m_R^T{m}_R;F_I:=m_I^T{m}_I$
  
• Third $$m_{20}:=\left(\frac{m_I}{\left|m_I\right|}-\frac{m_R}{\left|m_R\right|}\right)\sqrt{\frac{\left|m_I\right|\cdot \left|m_R\right|}{2}}$$
  
• 4. Calculate helper matrix S ₁ with algorithm (4): $${\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">S</figure-callout>}}_1=\text{<figure-callout id="1" label="R" filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png" state="{{state}}">R</figure-callout>}1\text{UD}\left(S=S_{R.k}.a=m_I\frac{\sqrt{F_{R0}}}{F_I}\right)$$
  
• 5. Calculate helper matrix S ₂ with algorithm (4): $${\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">S</figure-callout>}}_2=\text{<figure-callout id="1" label="R" filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png" state="{{state}}">R</figure-callout>}1\text{UD}\left(S={\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">S</figure-callout>}}_1.a={\mathrm{<figure-callout\; id="20"\; label="m"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0265.png"\; state="\{\{state\}\}">m</figure-callout>}}_{20}\frac{1}{\sqrt{F_I}}\right)$$
  
• 6. Calculate the helper matrix S ₃ with the algorithm (1): $${\mathrm{<figure-callout\; id="3"\; label="S"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0253.png"\; state="\{\{state\}\}">S</figure-callout>}}_3=\text{GCDD}\left(S={\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">S</figure-callout>}}_2.a=S_2{C}_{k+1}^T.R=0\right)$$
  
• 7. Calculate the result matrix S _{R, k + 1} with the algorithm (4): $$S_{R.k+1}=\text{R1UD}\left(S={\mathrm{<figure-callout\; id="3"\; label="S"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0253.png"\; state="\{\{state\}\}">S</figure-callout>}}_3.a=m_I\frac{R}{F_I}\right)$$
  
• 8. Calculate the result matrix S _{I, k + 1} and K _{k + 1} with the algorithm (1): $$\lfloor S_{I.\mathrm{<figure-callout\; id="1"\; label="k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">k\; </figure-callout>}+1}.K_{k+1}\rfloor =\text{GCDD}\left(S=S_{I.k}.a=m_I.R=0\text{tol}=\text{0}\right)$$
  

Equations (12b) - (12d) correspond to a Kalman filter-based restoration of the former state, where the matrix P _R may be singular. Therefore, according to various embodiments, the algorithm (1) can be applied. According to various embodiments, based on the algorithm (6), the square root of equations (12b) to (12d) may be formed as follows:

ALGORITHM 6:

1. 1. Input:
   • S _{R, k} - upper (n, n) triangular matrix
   • S _{I, k} - upper (n, n) triangular matrix
   • C _{k + 1} - (1, n) observation vector
   • R - scalar
2. 2. Calculate: $m_R:=S_{R.k}{C}_{k+1}^T$
   
3. 3. Calculate the result matrix S _{R, k + 1} and K _{k + 1} with the algorithm (1):
   • $\lfloor S_{R.k+1}.K_{k+1}\rfloor =\text{GCDD}\left(S=S_{R.k}.a=m_R.R.\text{tol}=\text{0}\right)$
     
4. 4. S _{I, k + 1} = S _{I, k}

• ■ die numerische Genauigkeit ist drastisch verbessert (~32 Stellen für doppelte Genauigkeit/die Konditionszahl von P_R , P_I ist auf ihre Quadratwurzel verringert),
• ■ der Algorithmus kann viele Zustände handhaben,
• ■ die Gesamtrechenkomplexität ist proportional zu n² (mit Ausnahme des Algorithmus (3), der maximal n-mal verwendet wird),
• ■ die Speicheranforderungen sind mit einem nicht modifizierten Kalman-Filter vergleichbar,
• ■ die S_I -Dreiecksmatrix verrät die Struktur der Mehrdeutigkeit (die Anzahl der von null verschiedenen Zeilen in S_I = die Anzahl der Freiheitsgrade),
• ■ die exakte Initialisierung der Fehlerkovarianzmatrix P mit ∞ ist möglich,
• ■ der Algorithmus ist der Modellmehrdeutigkeit und den Geräte-Dedikationen natürlich gewachsen.

According to various embodiments, for a Kalman-based filter or similar filter, a fast and stable square root algorithm is provided for restoring the previous state, as described above. The quick and stable square root algorithm for restoring the previous state may include one or more of the following:

• ■ The numerical accuracy is drastically improved (~ 32 digits for double precision / the condition number of P _R . P _I is reduced to its square root),
• ■ the algorithm can handle many states
• ■ the total computational complexity is proportional to n ² (except algorithm (3), which is used a maximum of n times),
• ■ memory requirements are comparable to a non-modified Kalman filter,
• ■ the S _I Triangular matrix reveals the structure of ambiguity (the number of non-zero rows in S _I = the number of degrees of freedom),
• ■ the exact initialization of the error covariance matrix P with ∞ is possible
• ■ The algorithm naturally grows with model ambiguity and device dedications.

According to various embodiments, the above-described fast and stable square root algorithm may be used to restore the past state in a run-to-run system component (eg, in semiconductor manufacturing), as previously described. According to various embodiments, when the production event degrades the state estimate (eg, due to a machining event, a maintenance event, etc.), data is added through equations (8a) through (8c), which is also referred to as a process update. When the production event has generated data (eg, in the case of a metrology event) that improves state estimation, the data is added by Equation (11a) through (12d), which is also referred to as a metrology update.

In diesem Fall ist σ_ERR die Standardabweichung für die Messdatenvorhersage. Ferner können gültige Messungen z. B. durch -L_y ≤ v₀ ≤ L_y mit 2 ≤ L_y ≤ 4 charakterisiert werden. Gemäß verschiedenen Ausführungsformen ist v₀ ein universelles Maß, das von dem Verwendungsfall und dem Übergangsverhalten (dem Einschwingen) des Schätzalgorithmus unabhängig ist. Falls v₀ L_y übersteigt, sind mehrere Reaktionen möglich, wie oben beschrieben worden ist.According to various embodiments, the accuracy information, e.g. As represented by the error standard deviation given by the equation (5) can be used as described above. The incoming measurement data y _{k + 1} can be calculated by calculating the normalized renewal of the ratio z. B. be validated according to the following: $$v_0=\frac{y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}}{{\sigma }_{eRR}},$$

In this case σ _{ERR is} the standard deviation for the measurement data prediction. Furthermore, valid measurements z. B. be characterized by -L _y ≤ v ≤ L _{0 y} with 2 ≤ L ≤ _y. 4 According to various embodiments, v _{0 is} a universal measure that is independent of the use case and the transient response (transient) of the estimation algorithm. If v ₀ exceeds L _y , several reactions are possible, as described above.

According to various embodiments, the data fusion circuit 108 or the process control circuit 100 include the above-described fast and stable square root algorithm for restoring the previous state. Therefore, the data merging circuit 108 have an important function in a control system, e.g. For example, to control discrete processes in the semiconductor industry and in other industries.

According to various embodiments, the data fusion circuit 108 or the process control circuit 100 include the above-described fast and stable prior art square root algorithm implemented as a component of a surveillance system (eg, the semiconductor industry or other industries). In semiconductor manufacturing, many processes must be monitored. The monitoring can z. For example, selecting an appropriate process model (eg, based on equation (1)) and a suitable measurement model (eg, based on equation (2)) for size, y which needs to be monitored and tuning the process model parameters by a statistical procedure using previous data (eg historical data) of the size y (eg, about the maximum likelihood estimate, which will be described in more detail below); Implementing equations (8a) to (8c) in the monitoring system (eg, via the data fusion circuit described herein 108 ), Calculating a state estimate x _k , P _{R, k} , P _{I, k} and calculating the normalized renewal, v ₀ (eg using equation 28) or any other error information of the state estimate and if new data y _{k +1} , validate 315 the new data 107d and / or validating 315 of the process model. The monitored process can, for. B. is validated, ie, behaves normally, in the case that -Ly ≤ v ₀ ≤ Ly with 1 ≤ Ly ≤ 5 or 2 ≤ Ly ≤ 4 applies.

As above z. B. with respect to the 6A to 6D has been described, a fast and numerically stable square root algorithm is provided for restoring the prior state of state information based on equations (11b) - (11d) and (12b) - (12d). The algorithm described above is based on a Kalman model, where the error covariance matrix is decomposed into a finite and an infinite part, which are decomposed into triangular square roots with lines that are exactly zero. The iteration equations (11b) - (11d) and (12b) - (12d) lead to a characteristic sequence of the state estimates x _k and the covariance vectors S _{R, k} , S _{I, k} k = 0, 1, ...

As above regarding the 6A to 6D has been described, the data fusion circuit 108 be configured, a distinguishing parameter (eg, F _I ) based on the state extraction matrix C and the error covariance matrix P provide. The distinguishing parameter can be compared with a discrimination limit, e.g. B. L _NUM , are compared.

As mentioned above, the Bierman algorithm is closely related to the Carlson algorithm and may have the same excellent numerical properties. He uses a generalizing Cholesky factorization according to the following: $$P=U^{T}DU,$$

$$P=U\cdot D\cdot U^T$$

In addition, according to various embodiments, algorithms 1, 2, 3, 4, 5, 6 may be reformulated with respect to equation (29). Instead of factoring (15) or (16), the following factorizations may be used to reformulate algorithms 1, 2, 3, 4, 5, 6 described above: $$P=S\cdot S^T$$

$$P=U\cdot D\cdot U^T$$

As an alternative to algorithm (1), the so-called Potter algorithm may be used to restore the previous state according to various embodiments. Like the Carlson algorithm, the Potter algorithm can be used for restoring the former state in the form of equation (17). An algorithm (7) for the previous state Potter restoration is shown by way of example below:

ALGORITHM 7:

1. 1. Input:
   • S - upper (n, n) triangular matrix
   • a - (n, 1) vector
   • R - Scalar
2. 2. Calculate: $F:=a^{T}a+{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2;\gamma :=1+\sqrt{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R</figure-callout>}}^2/F}$
   
3. 3. Calculate the result matrix: $\overline{S}=S-\frac{Sa{a}^T}{\gamma \cdot F}$
   

The algorithm (7) may be less precise than the Carlson algorithm. This can be z. In 6E to see the values of the distinguishing parameter F _I for the example 6B and 6D but calculated using the algorithm (7). The restoration of the previous state calculated with the algorithm (7) generates z. B. a numerical noise floor 604n from about 10 ^-25 , which is remarkably higher than that by the re 6D generalized Carlsen reconstruction of the previous state described generated numerical noise floor 602n from about 10 ^-33 's. Further, the memory space required for the covariance matrices is equal to that required by the previous state's Potter restoration. B. according to the algorithm ( 7 ) generated about four times higher compared to the generalized Carlsen reconstruction of the previous state described above. This is because the previous state's Potter restoration produces full matrices (rather than triangular matrices where only half of the matrix elements need to be stored) and does not generate lines that are exactly zero (which need not be stored).

ersetzt. Anstelle des Zustandsvektors x̂_kwird der Vektor ${\widehat{z}}_k:=I_k^{-1}{\widehat{x}}_k$

zusammen mit I_k ausgebreitet. In dem Fall keiner früheren Kenntnis können die Elemente von I_k mit 1/∞ = 0 initialisiert werden. Dies entspricht dem Initialisieren des Elements von P_k mit ∞. Dies bedeutet, dass das Informationsfilter eine Initialisierung mit unendlich und eine Modellmehrdeutigkeit natürlich handhaben kann.According to various embodiments, an information filter may be used to perform the functions of the data merge circuit 108 to implement. The information filter is a modification of the Kalman filter. In this modification, the error covariance matrix P _k is the inverse $I_k=P_k^{-1}$

replaced. Instead of the state vector x _k becomes the vector ${\widehat{z}}_k:=I_k^{-1}{\widehat{x}}_k$

spread out together with I _k . In the case of no prior knowledge, the elements of I _k can be initialized with 1 / ∞ = 0. This corresponds to initializing the element of P _k with ∞. This means that the information filter can naturally handle infinity initialization and model ambiguity.

berechnet werden. Die Rechenkomplexität des Berechnens des Inversen ist zu n³ proportional. Dies kann möglich sein, um eine Durchlauf-zu-Durchlauf-Steuerung zu implementieren, ist aber nicht für eine Durchlauf-zu-Durchlauf-Steuerung optimal, wo oft einige hundert Zustände verwendet werden und x̂_kund σ_ERR häufig berechnet werden müssen. Ferner kann das Informationsfilter außerdem durch das Radizieren der jeweiligen Gleichungen modifiziert werden.Whenever in this approach, an estimate x _{k of} the real state (or the estimation of linear state combinations â _k = C x _k ) and / or their corresponding estimation error standard deviation σ _{ERR are} required, the inverse $I_k^{-1}$

be calculated. The computational complexity of computing the inverse is proportional to n ³ . This may be possible to implement a pass-to-pass control, but is not optimal for pass-to-pass control, where often several hundred states are used and x _k and σ _ERR must be calculated frequently. Furthermore, the information filter may also be modified by the radicalizing of the respective equations.

Due to the model ambiguity, the error covariance matrix P _{k may be} singular, meaning that a simple inversion may not be possible. This corresponds to the fact that the individual states are not affected by the data unlike some state combinations â _k = C x _{k are} determined. In addition, the inversion algorithm should exploit a square root structure to preserve the excellent numerical properties. This can be z. By the following algorithm (8):

ALGORITHM 8:

• 1. Input:
  • Z - (n, n) triangular matrix; Square root of the information matrix, ie, P ^-1 = Z ^T Z
  • ẑ - converted (n, 1) state vector
  • C - (1, n) state extractor vector
• 2. Calculate the orthogonal column compressor T ^T (eg, using a singular value decomposition) $$Z{T}^T=Z\lfloor \begin{array}{cc}{T}_1^T& T_2^T\end{array}\rfloor =\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="1"\; label="Z"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">Z</figure-callout>}}_1& 0\end{array}\right].$$
  
  where the (n, r) matrix Z _{I is} column regular and r is the rank of Z
• 3. Calculate $\overline{C}{T}^T=C\lfloor \begin{array}{cc}{T}_1^T& T_2^T\end{array}\rfloor =:\left[\begin{array}{cc}{C}_1& C_2\end{array}\right]$
  
• 4. The estimation error of the linear state combination â _k = C x _k is only finite if C ₂ = 0. Do not use the state estimate if this condition is violated.
• 5. The Estimation of the State Combination α = C x and its finite error variance are given by: $$\widehat{\alpha }=C_1{\left(Z_1^T{Z}_1\right)}^{-1}\widehat{z};{\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2=C_1{\left(Z_1^T{Z}_1\right)}^{-1}{C}_1^T$$
  
  To ensure numerical robustness, the QR decomposition Z _I = WU and the back-replacement of the triangular system U ^T U are used to calculate the inverse $${\left(Z_1^T{Z}_1\right)}^{-1}=\left(U^T\underset{I}{\underset{\}}{W^{T}WU}}\right)={\left(U^{T}U\right)}^{-1}$$
  
  to calculate.

According to various embodiments, the error covariance matrix P may be decomposed into: $$P_k=L_k\left(D_{R.k}+\kappa \cdot D_{I.k}\right)L_k^T;\kappa \to \infty ,$$

During the iteration, the triangular matrix L _k and the diagonal matrices D _{R, k} and D _{I, k are} updated. In this alternative algorithm, the process update and the metrology update can not be separated. Furthermore, this alternative algorithm has a computational complexity of n ³ , which alternative algorithm is also numerically unstable.

The following is an algorithm that provides a robust distinction between the finite and infinite variance of the prediction error for a Kalman-based filter or any other similar filter based on e.g. To Equations (8a) to (12d) as described above. As described above, equation (10) may be a central decision point of the algorithm (see equations (11a) to (12d)) because the first case and the second case are differentiated (eg, demarcated) from each other need, such. B. based on a discrimination condition, such. B .: $$\left|F_{I.k+1}\right|>0$$

In diesem Fall muss der Schwellenwert ε (der außerdem als die Unterscheidungsgrenze bezeichnet wird) geeignet gewählt werden.Due to a limited numerical accuracy of any digital computer, the calculated value of F _I normally differs from the ideal value. Therefore, a modified discrimination condition can be used, such as. B .: $$\left|F_{I.k+1}\right|>\mathrm{\epsilon }$$

In this case, the threshold ε (which is also referred to as the discrimination limit) must be suitably selected.

Illustrate based on an example 6F . 6G and 6H that choosing the threshold may be relevant. Therefore, the observation vector C _{k + 1 is} varied. As examples shows 6F the distinguishing parameter | F _{I, k + 1} | for the observation vector C _{k + 1} = (1 1 1), shows 6G the distinguishing parameter | F _{I, k + 1} | for the observation vector C _{k + 1} = (10000 10000 10000) and shows 6H the distinguishing parameter | F _{I, k + 1} | for the observation vector C _{k + 1} = (0.0001 0.0001 0.0001). The area 610 in each figure, the interval illustrates where a threshold ε, which is safe between | F _{I, k + 1} | = 0 and | F _{I, k + 1} | > 0 is different, can be set. The threshold ε can be chosen so that it depends on the numerical noise floor 602n can be delimited, the threshold ε can z. B. one or more orders of magnitude above the numerical noise floor 602n be determined. However, there can not normally be a common overlap of the intervals for the three different observation vectors C _{k + 1} that are included in the 6F to 6H are illustrated.

According to various embodiments, the respective observation vector (e.g., in equation (11c)) may be z. B. be replaced by a modified observation vector. With C: = C _{k + 1} and | C ₀ | = 1, the modified observation vector can be formulated as follows: $$C={\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">C</figure-callout>}}_0\cdot \left|C\right|$$

mit P _I := P_I,k+1; P_I := P_I,k; F_I := F_I,k+1 In this case, | C | the length of the observation vector C. Substituting equation (30) into equation (11c) results in e.g. For example: $${\overline{P}}_I=P_I-\frac{P_I{C}_0^T{C}_0{P}_I}{F_I}\cdot \frac{{\left|C\right|}^2}{{\left|C\right|}^2}$$

With P _I : = P _{I, k + 1} ; P _I : = P _{I, k} ; F _I : = F _{I, k + 1}

As a result, the recursion of the infinite portion of the error covariance matrix P _{I may be} of the length of the observation vector C be independent and can by the direction of the observation vector C be completely determined. In contrast, the distinguishing parameter F _{I is} scaled by the squared length of C, which is e.g. B. is illustrated by the following equation: $$F_I=\underset{=:F_{I0}}{\underset{\}}{C_0{P}_I{C}_0^T}}\cdot {\left|C\right|}^2=F_{I0}\cdot {\left|C\right|}^2$$

$$\left|F_{I0}\right|\le \epsilon \to \text{Fall 2}$$

mit $$F_{I0}=C_0{P}_I{C}_0^T=\frac{F_I}{{\left|C\right|}^2}=\frac{F_I}{C{C}^T}$$

The distinguishing parameter F _I may be zero for any non-zero observation vector C if (and only if) F _{I0 is} zero. Therefore, the modified discriminating _parameter F _{I0 may be used} instead of the discriminating parameter F _I as follows to distinguish the first and second cases from each other: $$\left|F_{I0}\right|>\epsilon \to \text{<figure-callout id="1" label="case" filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png" state="{{state}}">case</figure-callout> 1}\text{.}$$

$$\left|F_{I0}\right|\le \epsilon \to \text{<figure-callout id="2" label="Case" filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png" state="{{state}}">Case</figure-callout> 2}$$

With $$F_{I0}=C_0{P}_I{C}_0^T=\frac{F_I}{{\left|C\right|}^2}=\frac{F_I}{C{C}^T}$$

The modified discriminator F _I0 may be more suitable for discriminating the cases than the non-modified discriminator F _I previously described, because the modified discriminator F _I0 is independent of the length of the observation vector C. This is z. In 6I illustrates that z. For example, the modified discriminator, F _I0 , for three pattern _{observation vectors} C _{k + 1} = (1 1 1), C _{k + 1} = (1 1 1) x 10000 and C _{k + 1} = (1 1 1) x 0.0001 shows the above regarding the 6F to 6H have been used. Even if the three pattern _{observation vectors} differ by several orders of magnitude, the values of the modified discriminator F _{10 are} aligned and highly comparable. Therefore, a common discrimination limit (see discriminator line 610d) can be used for all observation vectors. As has been shown above, this is not possible if the unmodified discriminator, F _I , is used.

The approach to modifying the discriminators F _I may also be applied to the above-described square root algorithms where the error covariance matrix is updated. This can be z. For example, be formulated with the following equations: $$F_{I0}=v_I^T{v}_I={\left|v_I\right|}^2;v_I=C_0{S}_I^T;$$

Furthermore, according to various embodiments, the error variance may be calculated for the prediction of arbitrary state combinations and outputs. The state prediction can z. Based on the following equation: $${\alpha }_{k+1}=\overline{C}{x}_{k+1}$$

$$F_I=\overline{C}{P}_{I,k}{\overline{C}}^{\text{T}}$$

In this case, α _{k + 1 is} the state combination of interest and is C the corresponding state extraction vector. The variance of the prediction error is given by the following equations: $${\mathrm{<figure-callout\; id="2"\; label="\sigma \; ERR"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\sigma \; </figure-callout>}}_{\text{ERR}}^{}=\{\begin{array}{cc}\overline{C}\left[P_{R.k}+Q_{K+1}{Q}_{k+1}^{\text{T}}\right]{\overline{C}}^{\text{T}}& \text{if}\left|F_I\right|=0\\ \infty & \text{if}\left|F_I\right|>0\end{array}$$

$$F_I=\overline{C}{P}_{I.k}{\overline{C}}^{\text{T}}$$

kann z. B. durch die Faktorisierung des Zustandsextraktionsvektors, z. B. in C = C ₀|C|, berechnet werden.The robust, modified discriminator $F_{I0}={\overline{C}}_0{P}_{I.k}{\overline{C}}_0^{\text{T}}$

can z. B. by the factorization of the state extraction vector, z. In C = C ₀ | C |, be calculated.

Furthermore, the output prediction z. Based on the following equation: $${\widehat{y}}_{k+1|k}=C_{k+1}{\widehat{x}}_k$$

$$F_I=C_{k+1}{P}_{I,k}{C}_{k+1}^{\text{T}}$$

In this case, ŷ _{k + 1 | k is} the estimate of y _{k + 1} based on the data until the time k and C _{k + 1 is} the corresponding observation vector. The estimation of the error variance is z. Given by the following equations: $${\mathrm{<figure-callout\; id="2"\; label="\sigma \; ERR"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\sigma \; </figure-callout>}}_{\text{ERR}}^{}=\{\begin{array}{cc}{C}_{k+1}{P}_{R.k}{C}_{k+1}^{\text{T}}+R_{k+1}{R}_{k+1}^{\text{T}}& \text{if}{F}_I=0\\ \infty & \text{if}{F}_I>0\end{array}$$

$$F_I=C_{k+1}{P}_{I.k}{C}_{k+1}^{\text{T}}$$

kann durch die Faktorisierung des Beobachtungsvektors, z. B. in C_k+1 = C₀|C_k+1|, entsprechend konstruiert werden. Es sind jedoch andere (z. B. komplexere) Vorhersageszenarios vorstellbar.The robust, modified discriminator $F_{I0}=C_0{P}_{I.k}{C}_0^{\text{T}}$

may be due to factorization of the observation vector, e.g. In C _{k + 1} = C ₀ | C _{k + 1} |, be constructed accordingly. However, other (eg more complex) forecasting scenarios are conceivable.

According to various embodiments, the fast and numerically stable square root algorithm may be used to update the state information and / or restore the prior state of state information in a component of a control system and / or to monitor discrete processes, as described above. In addition, a normalized observation vector (eg, a normalized state extraction vector) may be used to compute a robust, modified discriminator (eg, F _I0 ) to detect the infinite predictive variance of the Kalman-based filter. According to various embodiments, the discriminator may be stored for each metrology update to track the correct distinction of the first and second cases.

$$S_{R,k+1}^T{S}_{R,k+1}=A_{k+1}{S}_{R,k}^T{S}_{R,k}{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T$$

$$S_{I,k+1}^T{S}_{I,k+1}=A_{k+1}{S}_{I,k}^T{S}_{I,k}{A}_{k+1}^T$$

According to various embodiments, a fast and stable square root update algorithm for a Kalman filter is provided. The following is a fast and numerically stable square root algorithm for updating a Kalman based filter or any other similar filter based on e.g. To Equations (1), (1 *), (2), (2 *), (8) to (12) as described above. As described above, some typical problems of a pass-to-pass control can be solved by decomposing the error covariance matrices P into a finite and an infinite part. However, the square root algorithms described below as examples, which are based on the decomposed error covariance matrices P, may be applied in the same way to the non-decomposed error covariance matrices P, as also described herein. The numerical accuracy can be significantly improved if both parts are factored into square roots. This transforms the above-described equation (8a) into the following: $${\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+{\mathrm{<figure-callout\; id="1"\; label="B\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_{k+}$$

$$S_{R.k+1}^T{S}_{R.k+1}=A_{k+1}{S}_{R.k}^T{S}_{R.k}{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T$$

$$S_{I.k+1}^T{S}_{I.k+1}=A_{k+1}{S}_{I.k}^T{S}_{I.k}{A}_{k+1}^T$$

$$S_{I,k+1}^T{S}_{I,k+1}=S_{I,k}^T{S}_{I,k}$$

In a run-to-run application, no state transition matrix can be used, ie, (A _{k + 1} = I _n ). This may simplify equations (33), (34) for: $$S_{R.k+1}^T{S}_{R.k+1}=S_{R.k}^T{S}_{R.k}+Q_{k+1}{Q}_{k+1}^T$$

$$S_{I.k+1}^T{S}_{I.k+1}=S_{I.k}^T{S}_{I.k}$$

Equation (34a) may be trivial, while Equation (33a) may correspond to a square root update. However, the computational complexity of the conventional update algorithms is proportional to n ³ . The number n of states used in a pass-to-pass control is typically several hundred. Therefore, various embodiments will be described below that provide an update algorithm with reduced computational complexity.

• • Falls die Prozessrauschmatrix Q eine Diagonalmatrix ist, wird das Prozessrauschen nur in der Hauptdiagonale der Fehlerkovarianzmatrix P angesammelt.
• • Das Diagonalelement P(i, i) beeinflusst die Kalman-Iteration solange nicht, wie der Zustand x(i) in keine Metrologieaktualisierung einbezogen ist.
• • Die Anzahl der von null verschiedenen Elemente des Beobachtungsvektors C ist typischerweise viel kleiner als die Anzahl der Zustände (für eine Durchlauf-zu-Durchlauf-Steuerung: 2-4 gegen einige hundert).

According to various embodiments, the following observations were made on the sets of equations (3) and (4) described above:

• • If the process noise matrix Q is a diagonal matrix, the process noise becomes only in the main diagonal of the error covariance matrix P accumulated.
• • The diagonal element P (i, i) does not affect the Kalman iteration as long as the condition x (i) is not involved in any metrology update.
• The number of non-zero elements of the observation vector C is typically much smaller than the number of states (for a pass-to-run control: 2-4 vs. a few hundred).

• • Das akkumulierten Prozessrauschen kann von der Fehlerkovarianzmatrix P getrennt werden, siehe z. B. das Bezugszeichen 702.
• • Neues (z. B. ankommendes) Prozessrauschen 706 wird zu einer separaten Diagonalmatrix P₀ (z. B. für die Prozessaktualisierung) hinzugefügt, siehe z. B. das Bezugszeichen 704.
• • Wenn der Zustand x(i) in eine Metrologieaktualisierung einbezogen ist, wird das Prozessrauschen vor dem Aktualisieren der Fehlerkovarianzmatrix P von P₀(i, i) zu P(i, i) übertragen 700t.

These observations can be exploited to reduce the computational complexity of equation (33a). A modification may, for. B. as follows, see 7A :

• The accumulated process noise can be separated from the error covariance matrix P, see e.g. B. the reference number 702 ,
• • New (eg incoming) process noise 706 is added to a separate diagonal matrix P ₀ (eg for process update), see e.g. B. the reference number 704 ,
• If the state x (i) is involved in a metrology update, the process noise is transferred 700t before updating the error covariance matrix P from P ₀ (i, i) to P (i, i).

Such as In 7B is illustrated in a schematic view, the approach can be adapted to P = S ^T S and similar factorizations, as described above. In this case, the computational complexity of transmitting the process noise from P ₀ (i, i) to S ^T S is proportional to n ² , e.g. For example, to update the rank 1 as described above.

In diesem Fall ist P₀ eine (n, n)-Diagonalmatrix.This approach can be summarized in two algorithms, as described in more detail below. In general, the error covariance matrix P z. B. decomposed into the following: $$P=S^{T}S+{\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">P</figure-callout>}}_0$$

In this case, P _{0 is} an (n, n) diagonal matrix.

According to various embodiments, the algorithms described herein (eg, the algorithms (9), (10)) may be used to separate the process noise from the error covariance matrix P , such as As illustrated in equation (35), in the same way apply to the cases in which the error covariance matrix P into a finite part P _R and into an infinite part P _I , as has been described herein. In this case, the error covariance matrices P _R and P _I z. B. be decomposed as follows: $$P_R=S_R^T{S}_R+P_{0R}\text{or}{P}_I=S_I^T{S}_I+P_{0I},$$

können die Algorithmen (9), (10) wie folgt implementiert werden:With $S_{k+1}^T{S}_{k+1}+P_{0k+1}=S_k^T{S}_k+P_{0k}+Q_{k+1}{Q}_{k+1}^T$

For example, the algorithms (9), (10) can be implemented as follows:

ALGORITHM 9:

1. 1. Input:
   • S _k - upper (n, n) triangular matrix
   • P _{0, k} - (n, n) diagonal matrix
   • Q _{k + 1} - (n, n) diagonal matrix
2. 2. Calculate: $P_{0k+1}=P_{0k}+Q_{k+1}{Q}_{k+1}^T;S_{k+1}=S_k$
   

ALGORITHM 10:

1. 1. Input:
   • S _k - upper (n, n) triangular matrix
   • P _{0, k} - (n, n) diagonal matrix
   • C _{k + 1} - (1, n) observation vector
2. 2. Find all indices i with C _{k + 1} (i) ≠ 0 and P _{0, k} (i, i) ≠ 0
3. 3. For all indexes i found in step 2, perform the following steps:
   1. a. Initialize: a = (0 0 ... 0) ^T
   2. b. $a\left(i\right)=\sqrt{P_{0k}\left(i.i\right)}$
      
      put
   3. c. Use the algorithm (3) to modify S _k : S _k : = R 1UD (S _k, a)
   4. d. P _{0, k} (i, i) = 0
4. 4. Apply the square root algorithm for restoring the previous state to calculate the matrix S _{k + 1}

According to various embodiments with z. B. S ^T S = S ^T S + aa ^T may be an update of rank 1 z. Based on the algorithm (4) as described above.

According to various embodiments, a fast and numerically stable square root algorithm is provided for the process update, as described above. In this case, the computational complexity is reduced from the order n ³ to the order n ² when the number of non-zero elements in the observation vector C is much smaller than the number of states n. Furthermore, the memory requirements do not increase significantly because only n additional values need to be stored.

According to various embodiments, the fast and numerically stable square root algorithm may be used for process updating in a component of a control system and / or for monitoring discrete processes, as described above. According to various embodiments, the error covariance matrix may be decomposed into S ^T S + P ₀ or a similar sum. New process noise can be added to the diagonal matrix P ₀ . Process noise is transferred from P ₀ (i, i) (eg, only) to S ^T S when a state x (i) is included in the corresponding metrology update.

$$P=U^{T}DU+P_0,$$

$$P=UD{U}^T+P_0.$$

According to various embodiments, one or more alternative factorization may be used in a similar manner as described above. Instead of equation (35), one of the following alternative factorization may be used to reformulate algorithms (9) and / or (10): $$P=S{S}^T+{\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">P</figure-callout>}}_0.$$

$$P=U^{T}DU+{\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">P</figure-callout>}}_0.$$

$$P=UD{U}^T+P_0,$$

According to various embodiments, the respective factorization matrix S or U for the radiusing algorithm may be a (n, n) triangular matrix having a number of n rows and n columns, where n is a natural number, as illustrated above, see 6C , According to various embodiments, the respective factorization matrix used for the rendering algorithm may be an upper right triangular matrix, a lower left triangular matrix, a lower right triangular matrix, or a lower left triangular matrix. The error covariance matrix P is a symmetric (n, n) matrix.

If the error covariance matrix z. B. based on P = S ^T S or P = U ^T DU is factorized, as described above, the respective factorization matrix is S or U (and therefore also the error covariance matrix P) of rank R with a number of n rows, where they have a number of n minus R lines that are exactly zero. Further, in this case, for each of the n rows containing at least one element other than zero, the corresponding diagonal element of the respective factorization matrix S or U is not zero. Alternatively, if the error covariance matrix z. For example, based on P = SS ^T or P = UDU ^T , the respective factorization matrix is S or U (and therefore also the error covariance matrix P) of rank R with a number of n columns, where it is a number of n minus R columns which are exactly zero. Further, in this case, for each of the n columns containing at least one element other than zero, the corresponding diagonal element of the respective factorization matrix S or U is not zero. In both cases, according to various embodiments, the respective factorization matrix is a triangular matrix.

According to various embodiments, the rows that are exactly zero and / or the columns that are exactly zero allow a structural encoding of a reduction in rank. According to various embodiments, the data fusion circuit 108 be configured to generate a column or a zero row, if the rank of the respective factorization matrix S or U decreased by one. This allows the algorithm to easily identify a null row / column and start another procedure on nonzero rows. This can further increase the calculation unit. Further, this allows the algorithm to easily identify a singularity and to take into account the singularity during recovery of the previous state. The restoration of the previous state is z. B. formulated to preserve the properties of the matrices, z. To preserve the coding of the columns / rows which are exactly zero, accordingly.

According to various embodiments, here is provided a fast and numerically stable square root algorithm based on a block diagonal state transition matrix, as described in more detail below. Various algorithms are provided for a Kalman-based filter or any other similar filter based on equations (1), (1 *), (2), (2 *), as described above. In more detail, the state information of the process ( 1 ), (2) are estimated by two sets of equations that are performed completely independently, see the set of equations (3), (4), as described above.

$$S_{R,k+1}^T{S}_{R,k+1}+P_{\mathrm{0,}k+1}=A_{k+1}\left(S_{R,k}^T{S}_{R,k}+P_{\mathrm{0,}k}\right)A_{k+1}^T+Q_{k+1}{Q}_{k+1}^T,$$

$$S_{I,k+1}^T{S}_{I,k+1}=A_{k+1}\cdot S_{l,k}^T{S}_{l,k}\cdot A_{k+1}^T.$$

A fast and stable square root update algorithm for an infinity filter has been described above for the case where no state transition matrix, A, is used (e.g., for A _{k + 1} = I). This transforms the equations (8a, 8b, 8c) into: $${\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+{\mathrm{<figure-callout\; id="1"\; label="B\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">B\; </figure-callout>}}_{k+}.$$

$$S_{R.k+1}^T{S}_{R.k+1}+P_{0k+1}=A_{k+1}\left(S_{R.k}^T{S}_{R.k}+P_{0k}\right)A_{k+1}^T+Q_{k+1}{Q}_{k+1}^T.$$

$$S_{I.k+1}^T{S}_{I.k+1}=A_{k+1}\cdot S_{l.k}^T{S}_{l.k}\cdot A_{k+1}^T,$$

According to various embodiments, the process update algorithm may be configured to preserve the lines that are exactly zero, ensuring that all nonzero lines of zero have different diagonal elements, because the above-described corresponding square root reconstruction of the previous state may apply simultaneously should. According to various embodiments, the computational complexity of the algorithms described herein is less than proportional to n ³ .

It has been recognized that in practical applications the data fusion circuit 108 only a small number of states through the state transition matrix in the application of e.g. B. the equation (3) can be modified. Furthermore, one or more states may also share the same sub-transition matrix. There may be several (but only a few) sub-transition matrices.

According to various embodiments, the process update may include the following: First, according to various embodiments, the state transition matrix may be a block diagonal matrix 800 be organized, such. In 8th is illustrated. Second, only the nonunit blocks can 802 the block diagonal matrix 800 z. In a memory circuit, part of the process control circuit 100 or to the process control circuit 100 coupled can be stored. In this case all other diagonal blocks are the block diagonal matrix 800 1 × 1 unit matrices 804 that do not modify the condition. Third, the overall process update is now formulated as a series of sub-updates, with each sub-update being one of the nonunit blocks 802 processed. Fourth, the sub-update is on the rows and / or columns that make up the corresponding nonunit block 802 modified, restricted whenever possible. This significantly reduces the computational complexity because the number of these rows and / or columns is e.g. B. can be very small.

$${\overline{S}}_I^T{\overline{S}}_I=A\cdot S_I^T{S}_I\cdot A^T$$

According to various embodiments, the process update may be separated into two parts: first, a state transition, and second, an addition of process noise. For the second part, ie the completion of process noise, the method described above can be used. The first part, ie, the state transition, can be reformulated based on the following equations: $${\overline{S}}_R^T{\overline{S}}_R+{\overline{P}}_0=A\cdot \left(S_R^T{S}_R+P_0\right)\cdot A^T$$

$${\overline{S}}_I^T{\overline{S}}_I=A\cdot S_I^T{S}_I\cdot A^T$$

Gemäß verschiedenen Ausführungsformen kann die Gleichung (43) in der gleichen Weise für das Aktualisieren von S _k,R ^T S _k,R + P _0,k,R oder S _k,I ^T S _k,I+ P _0,k,I verwendet werden.By focusing on a single sub-update, equation (41) may be e.g. B. be formulated as equation (43) as follows: $${\overline{S}}^T\overline{S}+{\overline{P}}_0=\underset{A}{\underset{\}}{\left[\begin{array}{lllllll}1\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & 1\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & A_{*}\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & 1\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & 1\hfill \end{array}\right]}}\cdot \left(S^{T}S+\underset{P_0}{\underset{\}}{\left[\begin{array}{lllllll}{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">p</figure-callout>}}_1\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & p_{i-1}\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & P_{{}_{*}}^T{P}_{{}_{*}}\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & p_{j+1}\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & p_n\hfill \end{array}\right]}}\right)\cdot \underset{A^T}{\underset{\}}{\left[\begin{array}{lllllll}1\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & 1\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & A_{{}_{*}}^{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">T</figure-callout>}}\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & 1\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & 1\hfill \end{array}\right]}}$$

According to various embodiments, equation (43) may be used in the same way for updating S _{k, R} ^T S _{k, R} + P _{0, k, R} or S _{k, I} ^T S _{k, I} + P _{0, k, I} are used.

Ferner kann das Ausmultiplizieren der Gleichung (43) z. B. zu der Gleichung (45) wie folgt führen, die die Grundlage für den Aktualisierungsalgorithmus sein kann, wie im Folgenden ausführlicher beschrieben wird: $${\overline{S}}^T\overline{S}+{\overline{P}}_0=A\cdot S^{T}S\cdot A^T+\left[\begin{array}{lllllll}0\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & 0\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & A_{*}{P}_{*}^T{P}_{*}{A}_{*}^T\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & 0\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & 0\hfill \end{array}\right]+\underset{=:{\overline{P}}_0}{\underbrace{\left[\begin{array}{lllllll}{p}_1\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & p_{i-1}\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & 0\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & p_{j+1}\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & p_n\hfill \end{array}\right]}}$$

The corresponding nonunit block 802 is denoted by A *, while P * is the related part of the diagonal matrix P ₀ . He can z. For example, have the following structure: $$P_{*}=\left[\begin{array}{llll}\sqrt{p_i}\hfill & \hfill & \hfill & \hfill \\ \hfill & \sqrt{p_{i+1}}\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \sqrt{p_j}\hfill \end{array}\right]$$

Further, the multiplication of the equation (43) may be e.g. To equation (45), which may be the basis for the update algorithm, as described in more detail below: $${\overline{S}}^T\overline{S}+{\overline{P}}_0=A\cdot S^{T}S\cdot A^T+\left[\begin{array}{lllllll}0\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & 0\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & A_{*}{P}_{*}^T{P}_{*}{A}_{*}^{\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">T</figure-callout>}}\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & 0\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & 0\hfill \end{array}\right]+\underset{=:{\overline{P}}_0}{\underset{\}}{\left[\begin{array}{lllllll}{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">p</figure-callout>}}_1\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & p_{i-1}\hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & 0\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & p_{j+1}\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & p_n\hfill \end{array}\right]}}$$

Various embodiments described below are related to process update algorithms that include the state transition.

kann der Algorithmus (11) wie folgt formuliert werden:With the following equations: $${\overline{S}}_R^T{\overline{S}}_R+{\overline{P}}_0=A\cdot \left(S_R^T{S}_R+P_0\right)\cdot A^T\text{and}{\overline{S}}_I^T{\overline{S}}_I=A\cdot S_I^T{S}_I\cdot A^T$$

the algorithm (11) can be formulated as follows:

ALGORITHM 11:

• Input:
  • S _R - upper (n, n) triangular matrix
  • S _I - upper (n, n) triangular matrix
  • P ₀ - (n, n) diagonal matrix
  • A - block diagonal (n, n) state transition matrix
• For all (non-unit) block matrices A * From A do the following:
  • 1. The sentence I the row / column indexes, that of the current block matrix A * Match, determine. The smallest index in I denote by i.
  • 2. The columns I from S _R and S _I from the right with $A_{*}^T$
    
    multiply. This destroys the upper triangular shape of S _R and S _I. $$S_R=\left(1:n.I\right):=S_R\left(1:n.I\right)A_{*}^T$$
    
    $$S_I=\left(1:n.I\right):=S_I\left(1:n.I\right)A_{*}^T$$
    
  • 3. Restore the upper triangular shape of lines I of A by orthogonal row transformations using the algorithm (12). If A * is regular, the transformation can only be restricted to rows I. Otherwise, the transformation must be extended to lines i, i + 1, ..., n. In both cases, the columns become 1 . 2 , ..., i - 1 left unchanged.
    • Case 1 (A * is regular): $$S_R\left(I.i:n\right):=\text{MQR}\left[S_R\left(I.i:n\right)\right]$$
      
      $$S_I\left(I.i:n\right):=\text{MQR}\left[S_I\left(I.i:n\right)\right]$$
      
    • Case 2 (A * is singular): $$S_R\left(i:n.i:n\right):=\text{MQR}\left[S_R\left(i:n.i:n\right)\right]$$
      
      $$S_I\left(i:n.i:n\right):=\text{MQR}\left[S_I\left(i:n.i:n\right)\right]$$
      
  • 4. Find all indices in I with P ₀ (i, i) ≠ 0
  • 5. For everyone in the step 4 found indexes i do the following:
    • a. Initialize: a _R = a _I = (0 0 ... 0) ^T
    • b. $a_R\left(I\right)=a_{*}^i\sqrt{P_0\left(i.i\right)};a_I\left(I\right)=a_{*}^i$
      
      put up with $a_{*}^i$
      
      the i-th column of A * denotes
    • c. Use the algorithm (3) to modify S _R and S _I : $$S_R:=\text{R}1\text{UD}\left(S_R.a_R\right)$$
      
      $$S_I:=\text{R}1\text{UD}\left(S_I.a_I\right)$$
      
    • d. P ₀ (i, i) = 0

According to various embodiments, the (m, n) input matrix S becomes an upper triangular matrix S transformed. Furthermore, all lines that are exactly zero are preserved in the input matrix S during the update, with all nonzero lines of the input matrix S may have non-zero diagonal elements. Therefore, according to various embodiments, the algorithm (12) (also referred to as MQR) may include a modified Q, R decomposition as follows:

ALGORITHM 12:

1. 1. Input: S - (m, n) matrix
2. 2. The lines of S , which are exactly zero, detect and remove
3. 3. Use orthogonal row transformations (eg, a QR decomposition, Givens rotations, etc.) to S into an upper triangular matrix M to transform.
4. 4. The (m, n) result matrix S initialize with zeros
5. 5. For all lines r of M perform the following steps:
   1. a. Find the first non-zero element from the left and use the index of the element k describe
   2. b. If the line k is exactly zero, skip steps ce. Otherwise, the line r from M to the line k from S copy.
   3. c. If the line k from S is occupied, orthogonal row transformations (eg Givens rotations) to the row r of M and the line k from S apply to the element M (R, k ) to eliminate.
   4. d. All elements of the row r of M set to zero if | M (r, i) | <ε holds, wherein the threshold value ε z. B. the maximum norm of M could be.
   5. e. Repeat steps ae.

In this case, the algorithm (4) can be used to update the rank 1 used as described above.

According to various embodiments, the overall computational complexity of the algorithms (4), (11) and (12) is considerably smaller than the order n ³ , because the algorithms (normally) (instead of the total of n rows and / or columns) only the few rows and / or use columns that are the current nonunit block 802 correspond. Further, the algorithm preserves the lines that are exactly zero in the square root error covariance matrices. It also ensures that the nonzero lines in the square root error covariance matrices have non-zero diagonal elements.

According to various embodiments, the state transition matrix, A, may be used as a block diagonal matrix 800 be organized as described above. An updating algorithm may be used, as described above, (whenever possible) only the rows and / or columns of the nonunit block 802 processed.

The following is an algorithm for quickly calculating the exact gradient of the logarithmic probability function for a Kalman-based filter or any other similar filter based on e.g. Equations (1), (1 *), (2), (2 *) are provided as described above. As already described, the state information of the process ( 1 ) 2 ) are estimated by two sets of equations (eg, equations (3) and (4) or equations (8a) to (12d)) that are performed completely independently of each other.

According to various embodiments, the unknown state x _k may be estimated using a Kalman filter based algorithm. The respective Kalman filter-based algorithm may be used in the data fusion circuit 108 the process control circuit 100 be implemented. The state information and error information (provided, for example, based on the error covariance matrix) of the filter can accurately reflect the accumulation and degradation of the process information (the decrease and increase in the estimation error) as long as the process model parameters Q _k , R _k , B _k , D _k , C _k , A _k fit the process of real life.

In order to cope with the initialization problems and model ambiguity, as described above, it may be advantageous to decompose the error covariance matrix of the filter into a finite and an infinite portion. After each process or maintenance step, a process update can be performed. Using this method, the estimated state x _k and the decomposed covariance matrices P _{R, k} and P _{I, k} are updated based on the equations (8a), (8b) and (8c) as described above. Further, the metrology data may be added to the state by the metrology update based on the equations (9) to (12d) as described above. However, the determination of the process model parameters _Qk , _Rk , _Bk , _Dk may be a practical major aspect when applying Kalman-based filters to real production processes.

$$l_k=-\frac{1}{2}\left(\text{In}\left(F_{R,k}\right)+\frac{{\left(y_k-C_k{\widehat{x}}_{k-1}-D_k\right)}^2}{F_{R,k}}\right)$$

$$F_{R,k}=C_k{P}_{R,k-1}{C}_k^T+R_k^2$$

$$F_{I,k+1}:=C_{k+1}{P}_{I,k}{C}_{k+1}^T$$

In general, the production data may be an available source of information in practical situations. A powerful data-driven method to tune the Kalman filter parameters may be the maximum likelihood estimate. Probability is the likelihood that any given record will match the motel and its parameters. Unlike many other cases, the calculation of the (logarithmic) probability value for Kalman filter based models may be used, as described herein. According to various embodiments, the maximum likelihood estimate may be based on the following equations: $$\begin{array}{lll}L=\text{const}+{\displaystyle \underset{k=1}{\overset{N}{\Sigma }}{L}_k}\hfill & \text{With}\hfill & L_k=\{\begin{array}{lll}{l}_k\hfill & \text{if}\hfill & F_{I.k}=0\hfill \\ 0\hfill & \text{otherwise}\hfill & \hfill \end{array}\hfill \end{array}$$

$$l_k=-\frac{1}{2}\left(\text{In}\left(F_{R.k}\right)+\frac{{\left(y_k-C_k{\widehat{x}}_{k-1}-D_k\right)}^2}{F_{R.k}}\right)$$

$$F_{R.k}=C_k{P}_{R.k-1}{C}_k^T+{\mathrm{<figure-callout\; id="2"\; label="R\; k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}}_k^{}$$

$$F_{I.k+1}:=C_{k+1}{P}_{I.k}{C}_{k+1}^T$$

The goal of the maximum likelihood estimation may be the logarithmic probability value L to maximize. Because the dependence between the logarithmic probability and the process model parameters is non-linear, this is only possible through numerical optimization. Numerical optimization can be used to iteratively modify the process model parameters in the direction of the steepest increase in logarithmic probability. To the direction 900d of the steepest slope, according to various embodiments, the first-order partial derivatives of FIG logarithmic probability 900y (the gradient vector, the evaluation vector) for at least one model parameter 900x be calculated, such. In 9 is illustrated.

Several methods may be used to calculate the gradient vector of the logarithmic probability value L, such as: Finite differences, implicit finite differences, and the algorithm of exact evaluation (ES algorithm), as illustrated below.

z.B. $${\frac{\partial L}{\partial p_i}|}_{p=p*}\approx \frac{L\left(p_i^{*}+\Delta \right)-L\left(p_i^{*}-\Delta \right)}{\Delta }.$$

Around the gradient vector of the logarithmic probability value L Based on finite differences, according to various embodiments, the exact partial derivative of L with respect to the i-th tuning parameter by a difference quotient according to the following set of equations (50): $${\frac{\partial L}{\partial p_i}|}_{p=p*}\approx f\left[\Delta .L\left(p_i^{*}\right).L\left(p_i^{*}+\Delta \right).L\left(p_i^{*}+2\Delta \right).....L\left(p_i^{*}-\Delta \right).L\left(p_i^{*}-2\Delta \right)....\right]$$

eg $${\frac{\partial L}{\partial p_i}|}_{p=p*}\approx \frac{L\left(p_i^{*}+\Delta \right)-L\left(p_i^{*}-\Delta \right)}{\Delta },$$

In order to determine the full gradient vector for all p process model parameters, the logarithmic probability value L must be calculated for at least p + 1 slightly modified sets of process model parameters. The calculation of a single logarithmic probability value L, z. For example, using Equations (46) - (49), the Kalman filter equations (8a) - (12d) over the entire data set ( k = 1, 2, ..., N) to iterate. This means that the calculation of a p-dimensional gradient vector requires at least p + 1 full forward iterations of the Kalman filter. The procedure can be repeated for each optimization step.

Numerical optimization algorithms can be used to implicitly compute the finite differences of the optimized function (eg, via the conjugate gradient algorithm). This can be more efficient than explicitly calculating the full gradient for each optimization step. However, the effort can still be huge.

$$\frac{\partial l_k}{\partial p_i}=-\frac{1}{2}\text{tr}\left[\left(r_{k-1}{r}_{k-1}^T-N_{k-1}\right)\cdot \frac{\partial {\overline{Q}}_k}{\partial p_i}+\left(e_k{e}_k^T-E_k\right)\cdot \frac{\partial {\overline{R}}_k}{\partial p_i}\right]$$

$$\begin{array}{cc}{\overline{Q}}_k=Q_k{Q}_k^T;& {\overline{R}}_k=R_k^2\end{array}$$

According to various embodiments, an algorithm for computing the full gradient vector of the probability function with a single forward and backward iteration of the Kalman filter may be used. This algorithm can, for. For example, for the tuning process model parameters, e.g. Be applicable as the Rauschskalierungsmatrizen Q _k and R _k, according to a set of equations (50a) as follows: $${\frac{\partial L}{\partial p_i}|}_{p=p*}={\displaystyle \underset{k=1}{\overset{N}{\Sigma }}\frac{\partial L_k}{\partial p_i}}\text{With}\frac{\partial L_k}{\partial p_i}=\left\{\begin{array}{lll}\frac{\partial l_k}{\partial p_i}\hfill & \text{if}\hfill & F_{I.k}=0\hfill \\ 0\hfill & \text{otherwise}\hfill & \hfill \end{array}\right\}$$

$$\frac{\partial l_k}{\partial p_i}=-\frac{1}{2}\text{tr}\left[\left(r_{k-1}{r}_{k-1}^T-N_{k-1}\right)\cdot \frac{\partial {\overline{Q}}_k}{\partial p_i}+\left(e_k{e}_k^T-e_k\right)\cdot \frac{\partial {\overline{R}}_k}{\partial p_i}\right]$$

$$\begin{array}{cc}{\overline{Q}}_k=Q_k{Q}_k^T;& {\overline{R}}_k={\mathrm{<figure-callout\; id="2"\; label="R\; k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}}_k^{}\end{array}$$

According to various embodiments, the corresponding vectors r _k , e _k and matrices N _k , E _k can be calculated with the algorithm (13) as follows:

ALGORITHM 13:

• 1. x _k for k = 1, 2, ..., N using Equations (8a) - (12d). υ _k : = y _k - C _k x _k-1 - D _k , F _{R, k} , F _{I, k} and K _k for each k to save.
• 2. Initialization: r _N = 0; N _N = 0
• 3. for k = N, N - 1, ..., 1 $$e_k=-K_k^T{A}_{k+1}^T{r}_k+F_k^{-1}{\upsilon }_k.$$
  
  $$e_k=K_k^T{A}_{k+1}^T{N}_k{A}_{k+1}{K}_k+F_k^{-1}.$$
  
  $$r_{k-1}=A_{k+1}^T{r}_k+C_k^T{e}_k.$$
  
  $$N_{k-1}=\left(I-C_k^T{K}_k^T\right)A_{k+1}^T{N}_k{A}_{k+1}\left(I-C_k{K}_k\right)+C_k^T{F}_k^{-1}{C}_k$$
  
  With $$F_k^{-1}:=\{\begin{array}{ll}{F}_{R.k}^{-1}\hfill & \text{if}{F}_{l.k}=0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$
  
  The End

Illustratively, the algorithm (13) represents the backward iteration part of the ES algorithm. The algorithm for exact evaluation is not applicable if the measurement data for at least one point in time k are not available if any of the matrices Q _k and R _{k are} singular (which may be a very common situation in run-to-run applications) and / or for the tuning model parameters, e.g. B. for B _k and D _k .

According to various embodiments, here a modified algorithm (13) is provided to overcome the limitation of the conventional algorithm, wherein the modified algorithm provided herein, e.g. B. missing measurements has grown. According to various embodiments, the following algorithm (14) may be used to calculate the gradient vector of the logarithmic probability value L:

ALGORITHM 14:

Further, according to the set of equations (50a), according to various embodiments, the gradient calculation may be generalized to singular tuning matrices Q _k and R _k . Therefore, the following algorithm (15) e.g. Based on a modified set of equations (51).

ALGORITHM 15:

$${\frac{\partial L}{\partial p_i}|}_{p=p*}={\displaystyle \underset{k=1}{\overset{N}{\Sigma }}\frac{\partial L_k}{\partial p_i}\text{With}}\frac{\partial L_k}{\partial p_i}=\{\begin{array}{ll}\frac{\partial l_k}{\partial p_i}\hfill & \text{if}{F}_{l.k}=0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

$$\frac{\partial l_k}{\partial p_i}=-\frac{1}{2}\text{tr}\left[{\overline{Q}}_k^{+}\frac{\partial {\overline{Q}}_k}{\partial p_i}+\frac{\partial {\overline{Q}}_k^{+}}{\partial p_i}{\mathrm{\Omega }}_{Q.k}\right]-\frac{1}{2}\text{tr}\left[{\overline{R}}_k^{+}\frac{\partial {\overline{R}}_k}{\partial p_i}+\frac{\partial {\overline{R}}_k^{+}}{\partial p_i}{\mathrm{\Omega }}_{R.k}\right]$$

$${\mathrm{\Omega }}_{Q.k}={\overline{Q}}_k\left(r_{k-1}{r}_{r-1}^T-N_{k-1}\right){\overline{Q}}_k+{\overline{Q}}_k$$

$${\mathrm{\Omega }}_{R.k}={\overline{R}}_k\left(e_k{e}_k^T-e_k\right){\overline{R}}_k+{\overline{R}}_k$$

$${\overline{Q}}_k=Q_k{Q}_k^T;{\overline{R}}_k={\overline{R}}_{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2$$

bezeichnen das Moore-Penrose-Pseudoinverse von Q _k bzw. R _k. Bei der Durchlauf-zu-Durchlauf-Steuerung sind die Abstimmmatrizen (z. B. die Matrizen Q und R) oft einfache Diagonalmatrizen und können z. B. in der folgenden Form geschrieben werden: $$Q_k=\left[\begin{array}{cccc}\sqrt{{\displaystyle \sum _{i=1}{p}_i^2{w}_{Q\mathrm{,1,}i,k}^2}}& 0& 0& 0\\ 0& \sqrt{{\displaystyle \sum _{i=1}{p}_i^2{w}_{Q\mathrm{,2,}i,k}^2}}& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& \sqrt{{\displaystyle \sum _{i=1}{p}_i^2{w}_{Q,n,i,k}^2}}\end{array}\right]$$

$$R_k=\left[\begin{array}{cccc}\sqrt{{\displaystyle \sum _{i=1}{p}_i^2{w}_{R\mathrm{,1,}i,k}^2}}& 0& 0& 0\\ 0& \sqrt{{\displaystyle \sum _{i=1}{p}_i^2{w}_{R\mathrm{,2,}i,k}^2}}& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& \sqrt{{\displaystyle \sum _{i=1}{p}_i^2{w}_{R,m,i,k}^2}}\end{array}\right]$$

According to various embodiments, the corresponding vectors r _k , e _k and matrices N _k , E _k can be calculated with the algorithm (14) as described above. ${\overline{Q}}_k^{+}\text{and}{\overline{R}}_k^{+}$

denote the Moore-Penrose pseudoinverse of Q _k or R _k . In pass-to-pass control, the tuning matrices (eg, matrices Q and R) are often simple diagonal matrices and may be e.g. For example, written in the following form: $$Q_k=\left[\begin{array}{cccc}\sqrt{{\displaystyle \underset{i=1}{\Sigma }{p}_i^2{w}_{Q\mathrm{,1,}i.\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2}}& 0& 0& 0\\ 0& \sqrt{{\displaystyle \underset{i=1}{\Sigma }{p}_i^2{w}_{Q\mathrm{,\; 2,}i.\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2}}& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& \sqrt{{\displaystyle \underset{i=1}{\Sigma }{p}_i^2{w}_{Q.n.i.k}^2}}\end{array}\right]$$

$$R_k=\left[\begin{array}{cccc}\sqrt{{\displaystyle \underset{i=1}{\Sigma }{p}_i^2{w}_{R\mathrm{,1,}i.\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2}}& 0& 0& 0\\ 0& \sqrt{{\displaystyle \underset{i=1}{\Sigma }{p}_i^2{w}_{R\mathrm{,\; 2,}i.\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2}}& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& \sqrt{{\displaystyle \underset{i=1}{\Sigma }{p}_i^2{w}_{R.m.i.k}^2}}\end{array}\right]$$

In this case, p _i denotes the ith tuning parameter. The weighting factors w _{Q, r, i, k} and w _{R, r, i, k} can control the effect of p _i on the rth row / column of Q _k and R _k, respectively. The gradient calculation can be drastically simplified by taking advantage of the special structure of the matrices according to equations (52), (53). The result is z. In the algorithm (16) based on equation (54), as follows:

ALGORITHM 16:

In diesem Fall bezeichnen r_j,k und e_j,k den j-ten Elementvektor der Vektoren r_k bzw. e_k. N_jj,k und E_jj,k sind das (j, j)-Element (das j-te Hauptdiagonalelement) der Matrizen N_k und E_k. Ferner können die entsprechenden Vektoren r_k, e_k und Matrizen N_k, E_k mit dem Algorithmus (14) berechnet werden, wie oben beschrieben worden ist. $${\frac{\partial L}{\partial p_i}|}_{p=p*}={\displaystyle \underset{k=1}{\overset{N}{\Sigma }}{\displaystyle \underset{j=1}{\overset{n}{\Sigma }}\left[\left(r_{j.k-1}^2-N_{jj.k-1}\right)p_i^{*}{w}_{Q.j.i.k}^2\right]+}}{\displaystyle \underset{k=1}{\overset{N}{\Sigma }}{\displaystyle \underset{j=1}{\overset{m}{\Sigma }}\left[\left(e_{j.k}^2-e_{jj.k}\right)p_i^{*}{w}_{R.j.i.k}^2\right]}}$$

In this case, r _{j, k} and e _{j, k} denote the jth element vector of the vectors r _k and e _k, respectively. N _{jj, k} and E _{jj, k} are the (j, j) element (the jth major diagonal element) of the matrices N _k and E _k . Further, the corresponding vectors r _k , e _k and matrices N _k , E _k can be calculated with the algorithm (14) as described above.

According to various embodiments, the calculation of the exact gradients for the process model parameters, e.g. B. B _k and D _k , for calculating the first partial derivative of l _k in the equation (46) to be equivalent. According to various embodiments, the following equation (55) may be used: $$\frac{\partial l_k}{d{p}_i}=\frac{{\upsilon }_k}{F_{R.k}}\left[C_k{A}_k\frac{\partial {\widehat{x}}_{k-1}}{\partial p_i}+C_k\frac{\partial B_k}{\partial p_i}+\frac{\partial B{D}_k}{\partial p_i}\right]$$

kann rekursiv berechnet werden. Das Ergebnis ist z. B. in dem Algorithmus (17) wie folgt veranschaulicht:The unknown sensitivity vector $\frac{\partial {\widehat{x}}_{k-1}}{\partial p_i}=:z_{k-1}$

can be calculated recursively. The result is z. In the algorithm (17) as follows:

ALGORITHM 17:

According to various embodiments, the application of the algorithms (14) to (17), such. As described above, may not be limited to a filter based on equations (8a) - (12d). The algorithms can also be applied to the unmodified Kalman filter, such as. B. has been described above. The unmodified Kalman filter may be considered as a special case of the infinity filter (which is also referred to as a generalized Kalman filter) where the infinite part of the error covariance matrix P is zero (P _I = 0 and F _I = 0). The logarithmic probability value of the standard Kalman filter can be given by equations (46) - (49) with F _I = 0 and P _R = P. Thus, according to various embodiments, the algorithms (14), (15), (16) and (17) can be applied to the standard Kalman filter by setting F _I to zero.

According to various embodiments, the exact gradient vector can also be used in the case of missing measured values, in the case of singular noise matrices Q _k and R _k and for the tuning matrices B _k and D _k (eg for singular or regular tuning matrices) by the algorithms (14), ( 15), (16) and (17). Further, the algorithms provided herein allow calculating the exact gradient of the log likelihood function in the case of absence of measured values or singular or regular Abstimmmatrizen Q _k, R _k, B _k and D _k.

The following is a fast and stable square root algorithm for calculating the exact gradient of the logarithmic probability function for a Kalman-based filter or any other similar filter based on e.g. To Equations (1), (1 *), (2), (2 *), as described above. As has already been described, according to various embodiments, the state of the process ( 1 ) 2 ) are estimated by two sets of equations (eg, equations (3) and (4) or equations (8a) to (12d)) that are performed completely independently of each other.

According to various embodiments, the unknown state x _{k may be determined} using a Kalman filter based algorithm via the data fusion circuit 108 the process control circuit 100 is implemented, estimated. The filter state and the error covariance, the accumulation and the deterioration of the process information (the decrease and the increase of the estimation error) accurately reflect so long as the process model parameters Q _k, R _k, B _k, D _k, C _k, A _k to the process fit in real life. However, the determination of the process model parameters _Qk , _Rk , _Bk , _Dk can be a practical major consideration in applying Kalman filters to real production processes.

As described above, historical production data may be a convenient and available source of information in practical situations. Various algorithms for using a data driven method to tune the Kalman filter parameters based on the maximum likelihood estimation have already been described above. According to various embodiments, the maximum likelihood estimation may be based on the equations described above, cf. equations (46) through (49). In this case, P _{R, k} and P _{I, k are} the finite or infinite part of the error covariance matrix P the Kalman filter for the time index k , The goal of the maximum likelihood estimation may be to maximize the log likelihood value L as described above. To the direction 900d of the steepest slope, the first-order partial derivatives of the logarithmic probability 900y (the gradient vector, the evaluation vector) for a model parameter 900x be calculated, such. In 9 is illustrated.

• Fall 1: |F_I,k+1| = 0 (in der Praxis ≤ L_NUM) $$N_{k-1}=\left(I-C_k^T{K}_k^T\right)A_{k+1}^T{N}_k{A}_{k+1}\left(I-K_k{C}_k\right)+C_k^T{C}_k\cdot \frac{1}{F_{R,k}}$$
  
• Fall 2: |F_I,k+1| > 0 (in der Praxis > L_NUM) $$N_{k-1}=A_{k+1}^T{N}_k{\mathrm{<figure-callout\; id="1"\; label="A\; k\; +"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0258.png"\; state="\{\{state\}\}">A</figure-callout>}}_{k+}$$
  

One method of calculating the exact gradient vector that works for regular tuning matrices Q _k , R _k may require solving a recursive matrix equation of the following form:

• Case 1: | F _{I, k + 1} | = 0 (in practice ≤ L _NUM ) $$N_{k-1}=\left(I-C_k^T{K}_k^T\right)A_{k+1}^T{N}_k{A}_{k+1}\left(I-K_k{C}_k\right)+C_k^T{C}_k\cdot \frac{1}{F_{R.k}}$$
  
• Case 2: | F _{I, k + 1} | > 0 (in practice> L _NUM ) $$N_{k-1}=A_{k+1}^T{N}_k{A}_{k+1}$$
  

As has been described above, the method can be modified (generalized), to treat singular Abstimmmatrizen Q _k, R _k, B _k D _k. It is known that solving equations like equation (60) numerically can be a delicate task. In particular, the multiplication by (I-KC) and its transpose ("the restoration of the previous state") may be error prone. The problem is exacerbated by the fact that a typical run-to-run Kalman model may consist of several hundred states. According to various embodiments, the numerical stability of equations (60), (61) is significantly improved to more efficiently (eg, more accurately or more reliably) calculate the exact gradient vector for such a model, as described in more detail below.

According to various embodiments, a square root analog of these equations may be used to improve the numerical stability for solving equations (60), (61). Therefore, the matrix N _{k is} replaced by the following factorization or a similar factorization: $$N_k=:S_k^T{S}_k,$$

• Fall 1: |F_I| = 0 (in der Praxis ≤ L_NUM) $${\overline{S}}^T\overline{S}=\left(I-C^T{K}^T\right)A^T\cdot S^{T}S\cdot A\left(I-KC\right)+C^{T}C\cdot \frac{1}{F_R}$$
  
• Fall 2: |F_I| > 0 (in der Praxis > L_NUM) $${\overline{S}}^T\overline{S}=A^T\cdot S^{T}S\cdot A$$
  

In this case, instead of the matrix N _k, the square root matrix S _{k is} directly propagated. This strategy can also be applied to Kalman filtering as described above. The condition number of the square root matrix S _k (where the S _{k is} the Cholesky factor for the time index k are) much better than the condition number of the original matrix N _k . Substituting Equation (9) into Equations (7), (8) and omitting subscripts k and k + 1 for simplicity leads z. For example:

• Case 1: | F _I | = 0 (in practice ≤ L _NUM ) $${\overline{S}}^T\overline{S}=\left(I-C^T{K}^T\right)A^T\cdot S^{T}S\cdot A\left(I-KC\right)+C^{T}C\cdot \frac{1}{F_R}$$
  
• Case 2: | F _I | > 0 (in practice> L _NUM ) $${\overline{S}}^T\overline{S}=A^T\cdot S^{T}S\cdot A$$
  

In this case, the input matrix is S and the output matrix S Triangular matrices. The following algorithm (18) illustrates the square root procedure for equations (63), (64), where the algorithm (4) is the helper procedure used by the algorithm (18).

ALGORITHM 18:

• Input:
  • S - upper (n, n) triangle matrix
  • S - upper (n, n) triangle matrix
  • A - block diagonal (n, n) state transition matrix
  • C - (1, n) observation vector
  • K - (n, 1) -Kalman gain vector
  • F _R - finite part of the variance of renewal
  • F _I - infinite part of the variance of renewal
• 1. For all (non-unit) block matrices A * of A, do the following:
  1. a. Determine the set I of row / column indices corresponding to the current block matrix A *. Denote the smallest index in I with i.
  2. b. Multiply the columns I of S _R and S _I from the right by A *. This destroys the upper triangular shape of S. $$S\left(1:n.l\right):=S\left(1:n.l\right)A_{*}$$
     
  3. c. Restore the upper triangular shape of lines I of A by orthogonal row transformations using (e.g., QR decomposition, Givens rotations, etc.).
  The transformation can only be restricted to the lines I. The columns 1 . 2 , ..., i - 1 are left unchanged. $$S\left(I:i.n\right):=QS\left[S\left(I:i.n\right)\right]$$
  
• 2. The steps 3 - 7 skip if | F _I | > 0 (case 2)
• 3. Subtract the matrix S · K · C from the matrix S. This destroys the upper triangular form of S for all columns, which corresponds to the non-zero elements of vector C. $$S:=S\left(I-KC\right)=S-SKC$$
  
• 4. Select the leftmost column of the n _C faulty columns of S. Use the bottom element to eliminate all other column elements below the main diagonal with orthogonal row transformations (eg Givens rotations). This populates the below line of S.
• 5. Use the appropriate major diagonal elements to eliminate the artifact in the bottom row with orthogonal row transforms. The result is an upper triangular matrix with n _C - 1 perturbed columns.
• 6. The steps 4 and 5 Repeat until all disturbed columns are eliminated.
• 7. Use the algorithm (4) to add C ^T C · F ^-1 to S:

$$\overline{S}=\text{R1UD}\left(S.a=\frac{C^T}{\sqrt{F}}\right)$$

Such as In 10 is illustrated, destroys the subtracting of the matrix S · K · C (see the reference numeral 1002 ) from the matrix S (see the reference number 1004 ) the upper triangular shape 1000t the matrix S for all columns 1000c (see the reference number 1010 ), what the nonzero elements of the vector C equivalent.

According to various embodiments, the algorithm (4) may be for a rank update 1 used as described above.

According to various embodiments, similar to the algorithms described above, the algorithm (18) and the algorithm (4) can be applied both to the modified Kalman filter (which is also referred to as an infinity filter) and to the unmodified Kalman filter or any other similar Filters are applied. Because the Kalman filter can be a special case of an infinity filter, the algorithm (18) and algorithm (4) can be applied to the Kalman filter by taking the infinite part of the error covariance matrix P is set to zero (P _I = 0 and F _I = 0).

According to various embodiments, the fast and numerically stable square root algorithm for propagating equations (60), (61) as described herein may be used in a component of a control system and / or for monitoring discrete processes, as described above.

According to various embodiments, a fast and numerically stable square root algorithm is provided for the recursive matrix equations (60), (61).

The following is an algorithm for validating process models tuned to real data for a Kalman-based filter or any other similar filter based on e.g. To Equations (1), (1 *), (2), (2 *), as described above. As has already been described, according to various embodiments, the state of the process ( 1 ) 2 ) are estimated by two sets of equations (eg, equations (3) and (4) or equations (8a) to (12d)) that are performed completely independently of each other.

The unknown state x _k can be estimated using a Kalman filter-based algorithm, which uses the data fusion circuit 108 the process control circuit 100 is implemented. The filter state and the error covariance, the accumulation and the deterioration of the process information (the decrease and the increase of the estimation error) accurately reflect so long as the process model parameters Q _k, R _k, B _k, D _k, C _k, A _k to the process fit in real life. However, the determination of the process model parameters _Qk , _Rk , _Bk , _Dk can be a practical major consideration in applying Kalman filters to real production processes.

As described above, in general, historical production data in practical situations may be an available source of information. Above, various algorithms have already been described for using a data driven method to tune the Kalman filter parameters based on the maximum likelihood estimation. According to various embodiments, the maximum likelihood estimation may be based on equations (46) - (49) as described above.

Such as In 11 is illustrated (see. 9 ), the numerical optimization modifies the tuning parameters, p, iteratively in the direction of the steepest increase in logarithmic probability, as already described. Optimization stops, however, if the gradient (eg, about) is zero, see the reference numeral 1100Z ,

Therefore there is no guarantee that an optimization algorithm against a global maximum 1100g converges because the optimization is also at a local maximum 1100m can stop. Furthermore, in practical applications, it can not be ensured that the selected model structure matches the actual data. Both aspects can lead to a mismatch between the process model and the reality, which can cause a loss of yield and / or waste during manufacturing. Therefore, according to various embodiments, various algorithms are provided to reliably detect a mismatch of the model.

In diesem Fall entspricht k = 1,2, ..., N einem speziellen Zeitpunkt k. Der Optimierungsalgorithmus kann konfiguriert sein, alle Unterwahrscheinlichkeiten gleichzeitig zu maximieren, um die Gesamtwahrscheinlichkeit L zu maximieren. 12 veranschaulicht die l_k (siehe das Bezugszeichen 1200y) gegen das Verhältnis $F_{R,k}\text{/}{\upsilon }_k^2$

(siehe das Bezugszeichen 1200x) für einige feste Erneuerungen υ_k. In dem Fall, in dem das Verhältnis $F_{R,k}\text{/}{\upsilon }_k^2$

kleiner als 1,0 ist, ist der Wert F_R,k unterschätzt (siehe das Bezugszeichen 1200u), während in dem Fall, in dem das Verhältnis $F_{R,k}\text{/}{\upsilon }_k^2$

größer als 1,0 ist, der Wert F_R,k überschätzt ist (siehe das Bezugszeichen 1200o).Based on the average of the squared normalized updates or in a similar manner based on any suitable average taking into account multiple symmetrized normalized prediction errors, according to various embodiments, a model mismatch can be detected or the process model validated. The overall probability L z. In equation (46), the sum is off N individual sub-probabilities l _k . Each sub-probability can z. B. be formulated as follows: $$l_k=-\frac{1}{2}\left(\text{ln}\left(F_{R.k}\right)+\frac{{\left(y_k-C_k{\widehat{x}}_{k-1}-D_k\right)}^2}{F_{R.k}}\right)=-\frac{1}{2}\left(\text{ln}\left(F_{R.k}\right)+\frac{{\upsilon }_k^2}{F_{R.k}}\right)$$

In this case corresponds k = 1,2, ..., N a specific time k , The optimization algorithm may be configured to simultaneously maximize all sub-probabilities to maximize the overall L probability. 12 illustrates the l _k (see the reference numeral 1200y ) against the ratio $F_{R.k}\text{/}{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_k^{}$

(see the reference number 1200x ) for some fixed renewals υ _k . In the case where the ratio $F_{R.k}\text{/}{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_k^{}$

is less than 1.0, the value F _{R, k is} underestimated (see the reference numeral 1200u ), while in the case where the ratio $F_{R.k}\text{/}{\mathrm{<figure-callout\; id="2"\; label="\upsilon \; k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_k^{}$

is greater than 1.0, the value F _{R, k is} overestimated (see the reference numeral 1200o ).

auf, was zu ${\upsilon }_k^2\text{/}{F}_{R,k}=1$

äquivalent ist. Die letztere Bedingung kann z. B. als ${\upsilon }_{\mathrm{0,}k}^2=1$

mit $${\upsilon }_{\mathrm{0,}k}:=\{\begin{array}{cc}\frac{y_k-C_k{\widehat{x}}_{k-1}-D_k}{\sqrt{F_{R,k}}}=\frac{{\upsilon }_k}{\sqrt{F_{R,k}}}& \text{falls }{F}_{l,k}=0\\ 0& \text{sonst}\end{array}$$

umgeschrieben werden.The maximum 1200m occurs for $F_{R.k}={\mathrm{<figure-callout\; id="2"\; label="\upsilon \; k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\upsilon \; </figure-callout>}}_k^{}$

on, what to ${\upsilon }_{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2\text{/}{F}_{R.k}=1$

is equivalent. The latter condition can z. B. as ${\upsilon }_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2=1$

With $${\upsilon }_{0k}:=\{\begin{array}{cc}\frac{y_k-C_k{\widehat{x}}_{k-1}-D_k}{\sqrt{F_{R.k}}}=\frac{{\upsilon }_k}{\sqrt{F_{R.k}}}& \text{if}{F}_{l.k}=0\\ 0& \text{otherwise}\end{array}$$

be rewritten.

gemäß Gleichung (66) ein Hauptaspekt im Folgenden sein, wobei es als die normierte Erneuerung bezeichnet wird, wie bereits oben beschrieben worden ist. Das Maximum der logarithmischen Wahrscheinlichkeit L ist zum Verschieben aller quadrierten normierten Erneuerungen ${\upsilon }_{\mathrm{0,}k}^2,$

k = 1, 2, ..., N so nah wie möglich zu 1,0 äquivalent. Deshalb kann ein richtiges (globales) Maximum der Wahrscheinlichkeitsfunktion z. B. durch die folgende Gleichung charakterisiert werden: $$V_0=\frac{1}{N_0}{\displaystyle \sum _{k=1}^N{\upsilon }_{\mathrm{0,}k}^2\approx \mathrm{1,0.}}$$

In diesem Fall ist N₀ die Anzahl der Zeitpunkte k mit υ_0,k ≠ 0.According to various embodiments, the ratio ${\upsilon }_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2$

according to equation (66), a main aspect will be hereafter, being referred to as the normalized renewal, as already described above. The maximum of the logarithmic probability L is for shifting all squared normalized renewals ${\upsilon }_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2.$

k = 1, 2, ..., N as close as possible to 1.0 equivalent. Therefore, a correct (global) maximum of the probability function z. For example, be characterized by the following equation: $${\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">V</figure-callout>}}_0=\frac{1}{{\mathrm{<figure-callout\; id="0"\; label="N"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">N</figure-callout>}}_0}{\displaystyle \underset{k=1}{\overset{N}{\Sigma }}{\upsilon }_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2\approx \mathrm{1.0.}}$$

In this case, N _{0 is} the number of times k with υ _{0, k} ≠ 0th

so nah wie möglich zu 1,0. Dies ist nicht möglich, wenn das Modell nicht zu den Trainingsdaten passt. Deshalb kann gemäß verschiedenen Ausführungsformen die Fehlanpassung des Modells durch das Prüfen von |υ_0,k| oder ${\upsilon }_{\mathrm{0,}k}^2$

für Werte, die 1,0 z. B. signifikant übersteigen (z. B. > 3,0), geprüft werden.According to various embodiments, a mismatch of the model may be detected based on an observation of the normalized renewals, or the process model may be validated based on an observation of the normalized renewals, the normalized renewals Parameter that represents the normalized renewals, can or can, for. B. against the meter index k (also called the time index k is designated) are shown graphically. According to various embodiments, the optimization algorithm described above shifts all normalized updates ${\upsilon }_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2$

as close as possible to 1.0. This is not possible if the model does not match the training data. Therefore, according to various embodiments, the mismatch of the model can be checked by checking | υ _{0, k} | or ${\upsilon }_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2$

for values that are 1.0 z. Significantly higher (eg> 3.0).

13 illustrates an example of an observation of the normalized renewals, υ _{0, k} (see the reference numeral 1300y ) over time (see reference numeral 1300x ), the z. By the meter index k is represented. In the case where the normalized renewal 1.0 z. B. significantly exceeds, see the points 1300e , the process model may not be valid to describe the corresponding data.

According to various embodiments, the validation described above makes it possible to specifically analyze the problems of the process model at the points where the process model can not be valid (for example, for specific processes that are determined by the counter index k can be tracked).

According to various embodiments, a mismatch of the model may be detected based on an observation of the subprobability, or the process model may be validated based on an observation of the subprobability, the respective probability values or a parameter representing the subprobability may be e.g. B. against the meter index k graphically displayed. The total probability, L, in equation (46) decomposes into N individual sub-probabilities, l _k . The mismatch of the model may typically be limited to any fraction of the training data. The individual sub-probability values differ significantly in this case (a high probability of the data corresponding to the model and a low probability of the data mismatched to the model). Consequently, checking the sequence of individual sub-probabilities l _k with respect to significant wastes can be a powerful test.

14 illustrates an example of an observation of the individual sub-probability, l _k , (see the reference numeral 1400y ) over time (see reference numeral 1400x ), the z. By the meter index k is represented. In the case where the likelihood z. B. has a significant drop, see the points 1400e , the process model may not be valid to describe the corresponding data.

According to various embodiments, a mismatch of the model may be detected based on observation of the gradient signal, or the process model may be validated based on observation of the gradient signal, the gradient signal or a parameter representing the gradient signal may be e.g. B. against the meter index k graphically displayed. The numerical optimization algorithm can, for. For example, take advantage of the partial derivative of the total probability, see equation (46). Differentiating equation (46) with respect to the tuning parameter p _i shows that the overall gradient z. B. as follows N Subgradients decays: $${\frac{\partial L}{\partial p_i}|}_{p=p*}={\displaystyle \underset{k=1}{\overset{N}{\Sigma }}\frac{\partial L_k}{\partial p_i}\text{With}}\frac{\partial L_k}{\partial p_i}=\{\begin{array}{ll}\frac{\partial l_k}{\partial p_i}\hfill & \text{if}{F}_{l.k}=0\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

z.B. $${\frac{\partial l_k}{\partial p_i}|}_{p=p*}\approx \frac{l_k\left(p_i^{*}+\mathrm{\Delta }\right)-l_k\left(p_i^{*}-\mathrm{\Delta }\right)}{\mathrm{\Delta }}$$

Each subgradient ∂l _k / ∂p _i corresponds to a specific time k , The respective subgradient ∂l _k / ∂p _i can be determined by finite differences z. Be approximated as follows (see the set of equations (50a) described above): $${\frac{\partial l_k}{\partial p_i}|}_{p=p*}\approx f\left[\mathrm{\Delta }\mathrm{.}{l}_k\left(p_i^{*}\right).l_k\left(p_i^{*}+\mathrm{\Delta }\right).l_k\left(p_i^{*}+2\mathrm{\Delta }\right).....l_k\left(p_i^{*}\mathrm{-\Delta }\right).l_k\left(p_i^{*}-2\mathrm{\Delta }\right)....\right]$$

eg $${\frac{\partial l_k}{\partial p_i}|}_{p=p*}\approx \frac{l_k\left(p_i^{*}+\mathrm{\Delta }\right)-l_k\left(p_i^{*}-\mathrm{\Delta }\right)}{\mathrm{\Delta }}$$

in dem Fall, in dem p_i Q _k, R _k zugeordnet ist (siehe die Gleichung (51)) und $$\frac{\partial l_k}{\partial p_i}=\frac{{\upsilon }_k}{F_{R,k}}\left[C_k\left(A_k{z}_{k-1}+\frac{\partial B_k}{\partial p_i}\right)+\frac{\partial D_k}{\partial p_i}\right].$$

in dem Fall, in dem p_i B_k, D_k zugeordnet ist (siehe die Gleichung (55)).Using the methods described above, the subgradients ∂l _k / ∂p _{i can} also be z. For example, using the following equations: $$\frac{\partial l_k}{\partial p_i}=-\frac{1}{2}\text{tr}\left[{\overline{Q}}_k^{+}\frac{\partial {\overline{Q}}_k}{\partial p_i}+\frac{\partial {\overline{Q}}_k^{+}}{\partial p_i}{\mathrm{\Omega }}_{Q.k}\right]-\frac{1}{2}\text{tr}\left[{\overline{R}}_k^{+}\frac{\partial {\overline{R}}_k}{\partial p_i}+\frac{\partial {\overline{R}}_k^{+}}{\partial p_i}{\mathrm{\Omega }}_{R.k}\right]$$

in the case where p _i Q _k , R _{k is} assigned (see the equation (51)) and $$\frac{\partial l_k}{\partial p_i}=\frac{{\upsilon }_k}{F_{R.k}}\left[C_k\left(A_k{z}_{k-1}+\frac{\partial B_k}{\partial p_i}\right)+\frac{\partial D_k}{\partial p_i}\right],$$

in the case where p _i B _k , D _{k is} assigned (see equation (55)).

According to various embodiments, the gradient signal may be analyzed at the maximum likelihood value found by the numerical optimizer. The corresponding sub-gradient ∂l _k / ∂p _i shows how the tuning parameter p _i should be modified to the likelihood l _k for the time k to enlarge. Significant variations of ∂l _k / ∂p _i over time k can be a clear contradiction (the parameter p _i , for example, must be increased for some time intervals and reduced for others). Checking the gradient signals of all tuning parameters p _i may be a powerful mismatch test of the model, according to various embodiments. One possible consequence may be to divide conspicuous tuning parameters p _i into two or more parameters.

15 illustrates an example of observation of a gradient signal, ∂l _k / ∂p _i , (see reference numeral 1500y ) over time (see reference numeral 1500x ), the z. By the meter index k is represented. In the case where the gradient signal z. B. has a significant increase, see the point 1500e , the process model may not be valid to describe the corresponding data.

1. 1. Das Kalman-Filter über den vollen Satz der Trainingsdaten unter Verwendung der durch den Optimierer gefundenen Abstimmparameter laufen lassen. F_R,k und F_l,k und die Folge υ_k der Erneuerungen speichern.
2. 2. Die normierten Erneuerungen υ_0,k für alle k = 1, 2, ..., N mit der Gleichung (66) berechnen.
3. 3. V₀ mit der Gleichung (9) berechnen. Das Modell ablehnen, falls |V₀ - 1,0| irgendeinen Schwellenwert T₁ mit z. B. 0,1 ≤ T₁ ≤ 0,3 übersteigt.
4. 4. Die Folge υ_0,k gegen irgendeinen Schwellenwert T₂ prüfen. Das Modell ablehnen, falls |υ_0,k| T₂ für irgendein k mit z. B. 2 ≤ T₂ ≤ 5 übersteigt. Versuchen, das Modell für die Zeitintervalle, in denen T₂ überschritten wird, zu modifizieren.
5. 5. Die Folge der (Unter-) Wahrscheinlichkeiten l_k mit der Gleichung (65) berechnen. Das Modell ablehnen, falls l_k signifikante Abfälle zeigt. Versuchen, das Modell für die Zeitintervalle mit den Abfällen zu modifizieren.
6. 6. Die Folge der (Unter-) Wahrscheinlichkeiten für ∂l_k/∂p_i für alle Abstimmparameter p_i berechnen. Das Modell ablehnen, falls irgendeine Folge ∂l_kl/∂p_i signifikante Variationen zeigt. Versuchen, das Modell für die Zeitintervalle mit den Hauptvariationen zu modifizieren.

According to various embodiments, a mismatch of the model may be identified based on one or more of the procedures described herein. After an optimizer stops at any (possibly local) maximum, one or more of the following processes may be performed:

1. 1. Run the Kalman filter over the full set of training data using the tuning parameter found by the optimizer. Store F _{R, k} and F _{l, k} and the sequence υ _{k of} the renewals.
2. 2. The normalized renewals υ _{0, k} for all k = 1, 2, ..., N with the equation (66).
3. 3. Calculate V ₀ with equation (9). Reject the model if | V ₀ - 1,0 | any threshold T ₁ with z. B. 0.1 ≤ T ₁ ≤ 0.3 exceeds.
4. 4. Check the sequence υ _{0, k} against any threshold T ₂ . Reject the model if | υ _{0, k} | T ₂ for any one k with z. B. 2 ≤ T ₂ ≤ 5 exceeds. Try to modify the model for the time intervals in which T _{2 is} exceeded.
5. 5. Calculate the sequence of (sub) probabilities l _k using equation (65). Reject the model if l _k shows significant waste. Try to modify the model for the time intervals with the waste.
6. 6. Calculate the sequence of (sub) probabilities for ∂l _k / ∂p _i for all tuning parameters p _i . Reject the model if any sequence ∂l _k l / ∂p _i shows significant variations. Try to modify the model for the time intervals with the main variations.

According to various embodiments, such. For example, as described above, not only a mismatch of the model can be revealed, but also the times at which the mismatch occurs can be identified. This can be valuable information for improving the process model. The information of the logistical context (eg the process event data) for these times may be tested according to various embodiments. Therefore, a common pattern could be determined to find the root cause of the mismatch of the model.

Similar to the algorithms described above, the validation of the process model can be applied to both the modified Kalman filter and the unmodified Kalman filter or any other similar filter.

According to various embodiments, a mismatch of the process model may be detected based on at least one of the following parameters: renewal, υ _{0, k} , normalized renewal, the (sub) probability signal, or the (sub) gradient signal. Further, the time or a parameter representing the time may be used to locate the mismatch of the process model.

A user interface may, for example, according to various embodiments. B. in combination with any feedback device for accepting or rejecting the coordinated process models and / or measurement models as part of the process control circuit 100 provided or to the process control circuit 100 be coupled to indicate at least one of the validation measures / signals V ₀ , υ _{0, k} , l _k or ∂l _k / ∂p _i or their representatives.

As described above, the data fusion circuit may be configured to validate the process model and / or the measurement model based on an average value considering a plurality of symmetrized normalized prediction errors. Symmetrization can be useful to prevent positive and negative values from compensating each other. The symmetrized normalized prediction errors can be provided by applying a symmetric function to the normalized prediction errors. As an example, a root mean square (RMS) value may be used. According to various embodiments, the data fusion circuit 108 be configured to validate the process model and / or the measurement model based on a renewal signal, a probability signal and / or a probability gradient signal for one or more process model parameters of the process model.

According to various embodiments, the data fusion circuit may be configured to validate the process model and / or the measurement model on a set of training data. The set of training data may e.g. For example, historical data including any available process data (eg, context data and / or logistics data of the processing device) and measurement data may be included. The measured data can be z. B. be mapped to the process data or may be included in the process data.

Illustratively, the validation of the process model and / or the measurement model may include checking whether the available data set provided by the data merge circuit 108 the underlying process model and / or the measurement model according to which the data merge circuit 108 works, is described correctly.

According to various embodiments, reference measurement event data may be provided via the data fusion circuit 108 can be predicted and the newly acquired measurement event data compared with the predicted reference measurement event data. Based on this comparison, the process model and / or the measurement model can be validated, eg. If the newly acquired measurement event data may be correct, the process model and / or measurement model is not capable of predicting appropriate reference measurement event data, and therefore does not adequately describe the process. Furthermore, based on the comparison of the newly acquired measurement event data and the predicted reference measurement event data, the trade fair event and / or the corresponding process event can be validated, the newly acquired measurement event data can, for example, be validated. B. be excluded if the process model describes the process suitable.

The following is an algorithm for tracking, monitoring and / or debugging a Kalman-based filter or any other similar filter based on e.g. To Equations (1), (1 *), (2), (2 *), as described above. As has already been described, according to various embodiments, the state of the process ( 1 ) 2 ) are estimated by two sets of equations (eg, equations (3) and (4) or equations (8a) and (12d)) that are performed completely independently of each other.

According to various embodiments, the unknown state x _k may be estimated using a Kalman filter-based algorithm, which is provided via the data fusion circuit 108 of the Process control circuit 100 is implemented. In general, Kalman filter-based systems may lack transparency, which may lead to the problem that e.g. For example, process engineers may not be able to use these filters to control the processes in which a disruption could have serious consequences. This is caused by the high complexity of the algorithms in a Kalman-based filter compared to the conventional approaches. It can, for. For example, it may not be possible to retrace equations (8a) - (12d) using simple means such as paper and pencil. This can be an obstacle to acceptance of a Kalman filter-based controller by process engineers and production personnel.

One reason for the complexity (and performance) of a Kalman filter-based control system, as described herein, is the simultaneous updating of all states and all possible state combinations. A process update and a metrology update of any state combination may affect all other states. According to various embodiments, a tracking and debug tool is provided that is focused on one or more specific (user-defined) linear combinations of states, as described in more detail below.

During operation of the data fusion circuit 108 based on a Kalman filter problems can often arise for special linear state combinations. According to various embodiments, the analysis may be concentrated on this combination of states. Limiting the analysis to a linear combination of states dramatically reduces complexity according to various embodiments. The complicated behavior of the filter is illustrated by the more comprehensible behavior of the state combination.

The number of states and the potential linear state combinations of interest can typically be very high. It may not be possible to calculate and store the evolution of all states and state combinations of interest at runtime. According to various embodiments, various strategies are provided to reconstruct the complete development of the filter in retrospect. A first strategy may be to store all intermediate states of the filter, including the corresponding error covariance matrices. A second strategy may be to store the initial state of the filter and all Kalman matrices; and all intermediate states z. By reconstructing the respective equations (eg, equations (8a) - (12d)). A third strategy may be to store the intermediate states recurrently (including the corresponding error covariance matrices) and store all the Kalman matrices. In this case, any intermediate state may, if desired, be reconstructed by re-executing the respective equations for the time interval of interest beginning with a suitable (eg, the next) stored state.

The memory requirements can be huge for the first strategy. The second strategy may require re-executing the respective equations for the entire life of the filter, which may be e.g. B. may be one or more years, which leads to a huge computational effort. According to various embodiments, the third strategy for tracking, monitoring, and / or debugging the operation of the filter included in the data fusion circuit 108 the process control circuit 100 is used. The third strategy may be the most efficient in terms of both memory requirements and computational speed.

16A to 16C illustrate the third strategy as a schematic diagram of operation according to various embodiments. Starting from a corresponding state backup 1600b can be reconstructed by recalculation a desired intermediate state. The saved Kalman matrices 1600i can be used for recalculation. The saved Kalman matrices 1600i can have a corresponding identification value 1600V , which is also referred to as a pass-id, with the corresponding save state 1600b be correlated. The pass id can be used to specify data entry (eg, a process update, a measurement update, etc.) 1602 with the corresponding (eg next) fuse state to connect. Therefore, the pass ID defines the starting point for recalculating a respective state. In the following, three important analysis and / or debug usage cases are described with respect to the third strategy.

$${\sigma }_{\text{ERR}}^2=\{\begin{array}{ll}\overline{C}{P}_{R,k}{\overline{C}}^T\hfill & \text{falls }{\overline{F}}_I\le L_{NUM}\hfill \\ \infty \hfill & \text{falls }{\overline{F}}_I>L_{NUM}\hfill \end{array}$$

$${\overline{F}}_I=\overline{C}{P}_{l,k}{\overline{C}}^T$$

According to various embodiments, a first use case may be a detailed analysis of the evolution of a linear state combination C x of interest included. In this case, the linear combination of some states of interest C x _k and the corresponding estimation error are reconstructed. The analysis can z. Based on the following equations: $$a_k=\overline{C}{\widehat{x}}_k$$

$${\mathrm{<figure-callout\; id="2"\; label="\sigma \; ERR"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">\sigma \; </figure-callout>}}_{\text{ERR}}^{}=\{\begin{array}{ll}\overline{C}{P}_{R.k}{\overline{C}}^T\hfill & \text{if}{\overline{F}}_I\le L_{NUM}\hfill \\ \infty \hfill & \text{if}{\overline{F}}_I>L_{NUM}\hfill \end{array}$$

$${\overline{F}}_I=\overline{C}{P}_{l.k}{\overline{C}}^T$$

In this case, the extractor vector chooses C (see the reference number 1600e ) specify the states of interest, specifying the appropriate weighting factors, such as. In 16B is illustrated in a schematic diagram of the operation. The analysis can be generic and flexible because a generic extractor vector can be used, which can be adapted to all types of the Kalman filter-based model.

Therefore, according to various embodiments, the evolution of the linear state combination α _k can be investigated. In this way, one or more times k are identified, to which the linear state combination α _k and / or their estimation error change. These are the times when new information about the linear state combination α _{k is} added. Checking the appropriate logistical context information may reveal the root cause of the problems that might occur while operating the Kalman-based filter.

According to various embodiments, a second use case may be an identification of the prevailing readings for a linear state combination C x included. The Kalman filter / model can trust some readings more than other readings. It can be enlightening to identify the corresponding measured values that have a state combination α _k = C influence x _k of interest. According to various embodiments, the influence information (or in other words the impact) may be used to cross-check for the plausibility of the process model. In addition, the prevailing measurements can be extensively studied to determine possible anomalies. The Kalman gain K _k may be directly related to the effect of a measurement and may be used as a criterion for monitoring and the like. Because the absolute magnitude of the Kalman gain may be application-dependent, here, according to various embodiments, a normalized version of the Kalman gain is provided, which may be used as a measure of the impact.

$${\upsilon }_k=y_k-C_k{\widehat{x}}_{k-1}-D_k.$$

The Kalman gain, K _k , z. For example, be given by the following equations: $$K_k=\frac{P_{k-1}{C}_k^T}{C_k{P}_{k-1}{C}_k^T+R_k^T}=\frac{\text{e}\left[{\tilde{x}}_{k-1}\upsilon \right]}{\text{e}\left[{\upsilon }_k^2\right]}.$$

$${\upsilon }_k=y_k-C_k{\widehat{x}}_{k-1}-D_k,$$

In this case, υ _k and x _{k are} the renewal and the state estimation error, respectively. Multiplying Equation (75) with the state extractor vector C performs e.g. For example, to the following equation: $$\overline{C}{K}_k=\frac{\text{e}\left[\overline{C}{\tilde{x}}_{k-1}{\upsilon }_k\right]}{\text{e}\left[{\upsilon }_k^2\right]},$$

$$K_{\mathrm{0,}k}=\frac{\text{E}\left[\left(\overline{C}{\tilde{x}}_{k-1}\right)\cdot {\upsilon }_k\right]}{\sqrt{\text{E}\left[{\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\right]\cdot \text{E}\left[{\upsilon }_k^2\right]}}=\overline{C}{K}_k\cdot \sqrt{\frac{\text{E}\left[{\upsilon }_k^2\right]}{\text{E}\left[{\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\right]}}.$$

The numerator of equation (77) reveals z. B. that the Kalman gain, K _k , is controlled by a covariance of υ _k and x _k . The Kalman gain K _k is z. Zero, when the renewal υ _k and the state estimation error x _{k are} uncorrelated, while the absolute value of the Kalman gain K _{k is} 1.0 when the renewal υ _k and the state estimation error x _{k are} perfectly correlated. According to various embodiments, the correlation between the renewal υ _k and the state estimation error x _k may be used to obtain a normalized measure of the impact, K _{0, k} , z. B. according to the following equations: $$K_{0k}:=\text{corr}\left(\overline{C}{\tilde{x}}_{k-1}.{\upsilon }_k\right).$$

$$K_{0k}=\frac{\text{e}\left[\left(\overline{C}{\tilde{x}}_{k-1}\right)\cdot {\upsilon }_k\right]}{\sqrt{\text{e}\left[{\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\right]\cdot \text{e}\left[{\upsilon }_k^2\right]}}=\overline{C}{K}_k\cdot \sqrt{\frac{\text{e}\left[{\upsilon }_k^2\right]}{\text{e}\left[{\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\right]}},$$

$$\left|K_{\mathrm{0,}k}\right|:=\sqrt{\frac{\overline{C}{P}_{k-1}{\overline{C}}^T-\overline{C}{P}_k{\overline{C}}^T}{\overline{C}{P}_{k-1}{\overline{C}}^T}}$$

According to various embodiments, there may be one or more alternative definitions of the normalized measure of impact, K _{0, k} , that may be equivalent to equation (79), such as. For example, the following definition: $$\left|K_{0k}\right|:=\sqrt{\frac{\text{e}\left[{\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\right]-\text{e}\left[{\left(\overline{C}{\tilde{x}}_k\right)}^2\right]}{\text{e}\left[{\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\right]}}$$

$$\left|K_{0k}\right|:=\sqrt{\frac{\overline{C}{P}_{k-1}{\overline{C}}^T-\overline{C}{P}_k{\overline{C}}^T}{\overline{C}{P}_{k-1}{\overline{C}}^T}}$$

in der Gleichung (80) ist die Varianz des Schätzfehlers für die Zustandskombination α_k = Cx̂_k. Dies führt zu einer weiteren intuitiven Interpretation des normierten Maßes der Auswirkung, K_0,k, nämlich, dass die quadrierte Verstärkung $K_{\mathrm{0,}k}^2$

die relative Verringerung der Varianz des Schätzfehlers von α_k-1, aufgrund eines Messwerts y_k ist.The expected value $\text{e}\lfloor {\left(\overline{C}{\tilde{x}}_{k-1}\right)}^2\rfloor$

in equation (80), the variance of the estimation error for the state combination is α _k = C x _k . This leads to a further intuitive interpretation of the normalized measure of the impact, K _{0, k} , namely that of the squared amplification $K_{0\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">k</figure-callout>}}^2$

is the relative reduction of the variance of the estimation error of α _k-1 due to a measurement value y _k .

According to various embodiments, the algorithm (19) may be used to calculate the normalized measure K _{0, k of} the effect in the case where a decomposed error covariance matrix is used for the state estimation, as described herein, as described in more detail below becomes. If a conventional Kalman filter is used, F _I and F _I in the algorithm (19) are set to zero. Further, as will be described in more detail below, the algorithm (20) may be used in the case where a factorized error covariance matrix is used for the state estimation, as described above.

ALGORITHM 19:

ALGORITHM 20:

According to various embodiments, a third use case may include an analysis of a predicted value of interest. The algorithms based on a Kalman filter, as described above, can predict the process state to calculate the optimal tuning parameters for the next materials (the next batch, the next wafer, or the next batch). Consequently, the origin of the predicted value may be relevant, especially if the value is suspect. According to various states, the current state x _{k of} the Kalman filter based algorithms can be decomposed into a weighted sum of all previous measurements and a constant. This can be z. B. expressed as follows: $${\widehat{x}}_k=\text{const}+{\displaystyle \underset{i=1}{\overset{k}{\Sigma }}{w}_i{y}_i}$$

In this case, the weights w _{i of} the sum may be calculated using the algorithm (21), as described in more detail below. Multiplying z. For example, equation (82) from the left with some state extraction vector C leads to: $$\overline{C}{\widehat{x}}_k=\text{const}+{\displaystyle \underset{i=1}{\overset{k}{\Sigma }}{\overline{w}}_i{\overline{y}}_i},$$

In this case, the left side of the equation (83) is the predicted value α _k : = C x _k , if C the corresponding state extraction vector is.

$$\mathrm{\Delta }{\alpha }_k={\displaystyle \sum _{i=1}^k{\overline{w}}_i}\mathrm{\Delta }{y}_i$$

The weights w _i are scalars that z. B. can be displayed graphically against the time. It also reveals the weight w _{i is} the effect of a measured value y _i on the prediction. This can be valuable information for the operation of the Kalman-based filter. First, all metrics can be identified with a significant impact and checked for anomalies. Second, they can be used to quickly correct the prediction if erroneous readings are found. The equation (83) shows z. For example, the prediction correction is a simple weighted sum of the measurement corrections, ie: $${\alpha }_{CORR.k}={\alpha }_k+\mathrm{\Delta }{\alpha }_k$$

$$\mathrm{\Delta }{\alpha }_k={\displaystyle \underset{i=1}{\overset{k}{\Sigma }}{\overline{w}}_i}\mathrm{\Delta }{y}_i$$

The scalar weights w _i can use the recurring stored states as above 16A has been described, and z. B. the algorithm (22) are calculated. The complete procedure is z. In 16C in a schematic diagram 1600w of operation according to various embodiments.

ALGORITHM 21:

ALGORITHM 22:

According to various embodiments, the algorithm for tracking, monitoring, and / or debugging a Kalman-based filter may be used in a component of a control system and / or for monitoring discrete processes, as described above. According to various embodiments, the intermediate states and / or the Kalman matrices may be stored. Based on these stored intermediate states and / or the stored Kalman matrices, different tracking and / or debug sizes, e.g. B. estimating α _k, the estimation error σ _{ERR, k,} the measured value Impact K _{0, k,} the measured value weight w _i , etc. calculated and / or graphed. This allows z. For example, focus on special user-defined linear combinations of states.

According to various embodiments, the data fusion circuit 108 be configured to have multiple backup state information and multiple backup model parameters (eg, the model parameters for the time k . k - 1, k - 2) to reconstruct at least one state information to be monitored. Illustratively, historical model parameters may be used to reconstruct at least one state information to be monitored. The reconstruction of the at least one state information to be monitored may be efficient, as described above, when the recalculation starts from one of the stored multiple backup state information that is proximate in time.

According to various embodiments, the data fusion circuit 108 be configured to decompose the state information into a weighted sum considering a plurality of previous measurement event data, multiple backup model parameters (eg, process model parameters and measurement model parameters), and some appropriate backup states.

According to various embodiments, a method is provided to reduce the size of Kalman filter datasets. Therefore, an algorithm for a Kalman-based filter or any other similar filter based on e.g. To Equations (1), (1 *), (2), (2 *), as described above. As has already been described, according to various embodiments, the state of the process ( 1 ) 2 ) are estimated by two sets of equations (eg, equations (3) and (4) or equations (8a) to (12d)) that are performed completely independently of each other.

According to various embodiments, the unknown state x _k may be estimated using a Kalman filter-based algorithm, which is provided via the data fusion circuit 108 the process control circuit 100 is implemented. The filter state and the error covariance, the accumulation and the deterioration of the process information (the decrease and the increase of the estimation error) reflect long as precise as the process model parameters Q _k, R _k, B _k, D _k, C _k, A _k to the process in fit real life.

As already described, historical production data in practical situations may be an available source of information. Calculating the Kalman filter parameters from historical production data can be an expensive and time consuming task. The historical dataset may contain several hundred thousand individual materials, e.g. B. produced wafers, and their corresponding process event data and, if available, their corresponding measurement event data. The process model tuning procedure may be based on numerical optimization, which may require running the Kalman filter (and optionally a gradient calculation) for each optimization step over the entire data set. Therefore, the model design process may be compromised because the high overhead limits the number of model variations that can be tested.

In semiconductor manufacturing z. For example, only a fraction of the processed wafers are measured (eg, typically 20% or less). Therefore, according to various embodiments, the process equations (1), (1 *) can be combined in the case of missing and / or non-validated measured values. As an example, measurements for times k ₀ and k be available while for the intermediate time points k ₀ + 1 k ₀ + 2, ..., k - 2, k - 1 missing. The insertion of the corresponding process equations into each other for z. To the following unified equation (100), which is similar to equation (1): $$\begin{array}{c}{x}_k=A_k{A}_{k-1}\cdots A_{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+1}{x}_{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">k</figure-callout>}}_0}\\ +{\displaystyle \underset{i=k_0}{\overset{k-1}{\Sigma }}\left[A_k{A}_{k-1}\cdots A_{i+1}{Q}_i{\xi }_i\right]+Q_k{\xi }_k}\\ +{\displaystyle \underset{i=k_0}{\overset{k-1}{\Sigma }}\left[A_k{A}_{k-1}\cdots A_{i+1}{B}_i\right]+B_k}\end{array}$$

mit $${\overline{A}}_k=A_k{A}_{k-1}\cdots A_{k_0+1}$$

$${\overline{B}}_k={\displaystyle \sum _{i=k_0}^{k-1}\left[A_k{A}_{k-1}\cdots A_{i+1}{B}_i\right]+B_k}$$

$$\text{E}\left[{\overline{Q}}_k{\overline{\xi }}_k{\overline{\xi }}_k^T{\overline{Q}}_k^T\right]={\overline{Q}}_k{\overline{Q}}_k^T={\displaystyle \sum _{i=k_0}^{k-1}\left[A_k{A}_{k-1}\cdots A_{i+1}{Q}_i{Q}_i^T{A}_{i+1}^T\cdots A_{k-1}^T{A}_k^T\right]+Q_k{Q}_k^T}$$

Equation (100) may indeed be a process equation containing the state information x _{k 0} linked directly to the status information x _k . This process equation can, for. B. be reworded as follows: $$x_k={\overline{A}}_k{x}_{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">k</figure-callout>}}_0}+{\overline{Q}}_k{\overline{\xi }}_k+{\overline{B}}_k$$

With $${\overline{A}}_k=A_k{A}_{k-1}\cdots A_{{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">k</figure-callout>}}_0+1}$$

$${\overline{B}}_k={\displaystyle \underset{i=k_0}{\overset{k-1}{\Sigma }}\left[A_k{A}_{k-1}\cdots A_{i+1}{B}_i\right]+B_k}$$

$$\text{e}\left[{\overline{Q}}_k{\overline{\xi }}_k{\overline{\xi }}_k^T{\overline{Q}}_k^T\right]={\overline{Q}}_k{\overline{Q}}_k^T={\displaystyle \underset{i=k_0}{\overset{k-1}{\Sigma }}\left[A_k{A}_{k-1}\cdots A_{i+1}{Q}_i{Q}_i^T{A}_{i+1}^T\cdots A_{k-1}^T{A}_k^T\right]+Q_k{Q}_k^T}$$

The unified equations (100), (100 *) are among the original ones k - k ₀ Interprocess equations equivalent. According to various embodiments, the replacement of all intermediate process equations by the merged equations may drastically shrink the data set (eg, by 80% and more depending on the percentage of missing measurements).

verwenden, die z. B. direkt durch die Gleichung (103) gegeben ist. In dem Fall, dass z. B. ein Quadratwurzelfilter verwendet wird, wie oben beschrieben worden ist, kann die zugrundeliegende Matrix Q _k entsprechend, z. B. unter Verwendung des Algorithmus (23), berechnet werden. Dies kann z. B. unter Verwendung des folgenden Algorithmus ausgeführt werden:According to various embodiments, the combined Kalman matrices A _k and B _k can be calculated using equations (101) and (102), respectively. The Kalman filter can be the squared matrix ${\overline{Q}}_k{\overline{Q}}_k^T$

use, the z. B. is given directly by the equation (103). In the case that z. For example, as a square root filter is used, as described above, the underlying matrix Q _k accordingly, z. By using the algorithm (23). This can be z. For example, using the following algorithm:

ALGORITHM 23:

• 1. $M:={\overline{Q}}_k{\overline{Q}}_k^T$
  
  using equation (7)
• 2. M using z. As the Cholesky algorithm or the Singularwertzerlegung into the Cholesky factors M = SS ^T decompose
• 3. The result matrix Q _k is given by the Cholesky factor S.

According to various embodiments, all process noise matrices Q _k and all state transition matrices A _k z. B. be in practical applications diagonal matrices. In this case, the calculation is significantly simplified, as illustrated by the algorithm (24) as follows.

ALGORITHM 24:

• 1. $M:={\overline{Q}}_k{\overline{Q}}_k^T$
  
  using equation (103)
• 2. The element Q _k (i, i) of the result diagonal matrix Q _k is through $\sqrt{M\left(i.i\right)}$
  
  given

According to various embodiments, the algorithm for reducing the size of a data set of a Kalman-based filter provided herein may be used in a component of a control system and / or for monitoring discrete processes, as described above. According to various embodiments, the size of one or more tuning data sets may be shrunk by merging intermediate process equations that do not take into account measured values. In other words, a consecutive sequence of process updates may be merged in the event that none of the process updates of the consecutive sequence is correlated with a measurement that needs to be taken into account.

According to various embodiments, the data fusion circuit 108 be configured to provide a unified model parameter (eg, a process model parameter) for each one or more of the model parameters, respectively. In this case, the unified model parameter represents several model parameters of the corresponding events for which unification results in the same result as for the unrelated events. As an example, multiple process model parameters may be merged according to a (eg, consecutive) sequence of process events if no measurement events are to be considered for the sequence of process events.

The following provides an algorithm for modeling an interruption or some other disturbance in the complex and dynamic process. As has already been described, a Kalman-based filter or any other similar filter based on e.g. On the equations (1), (1 *), (2), (2 *) for controlling a production process. The state of the process ( 1 ) 2 ) can be represented by two sets of equations (eg equations (3) and (4) or equations (8a) to (12d)) estimated to be performed completely independently of each other. The normalized renewals v ₀ can z. Calculated based on the following equation: $${\mathrm{<figure-callout\; id="0"\; label="v"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">v</figure-callout>}}_0=\frac{y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}}{\sqrt{C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T}}$$

As described above, the normalized updates or any other suitable parameter representing the accuracy of the state estimate may be monitored. If the used process model (and / or measurement model) and the real process match, it can be very unlikely that the normalized renewals will differ significantly from 3, eg. For example, that | v ₀ | is greater than 3. According to various embodiments, a mismatch of the model can be assumed, if | v ₀ | exceeds any given threshold value L _y with 2 ≤ L _y ≤ 4. In other words, the process model and / or the measurement model can be based on the normalized renewals | v ₀ | or any other suitable parameter representing the accuracy of the state estimate.

The following provides a method to validate the process model and / or the measurement model. According to various embodiments, a check for mismatch of the model may be performed when new measurement data is fed to the estimation algorithm. A mismatch of the model can z. After a maintenance event for a machining tool of the machining assembly 104 or due to an erratic operation of a machining tool of the machining assembly 104 and the like, which is referred to herein as an interrupt. Such as In 17A is illustrated in a schematic diagram, a standard operation 1700s the filter at a given time 1700T be interrupted 1700d. During standard operation 1700s can a corresponding process model 1700m be used for the operation of the filter.

In the case where a mismatch of the model is detected, according to various embodiments, a set of alternative models 1702 defined, such. In 17B is illustrated. Each of the alternative models 1702 may correspond to a possible major cause of the interruption. The alternative models 1702 and the process model 1700m that have been used so far may be in a model library 1712 to be collected. Further, according to various embodiments, the state estimate for each model in the model library 1712 be calculated. If necessary or helpful, the models can be updated to the current data, e.g. With a maximum likelihood estimate, as described above. Further, according to various embodiments, the performance of the various models of the model library 1712 be compared, for. By comparing the mean square prediction error, the likelihood (eg, the likelihood that the model matches the actual data), or an information criterion (eg, an akaike information criterion, a Bayesian information criterion), and the like. This may also be referred to as testing or validating the process models. Further, according to various embodiments, the best performing model may be selected based on the result of the testing 1704 , The procedure may be seen as testing various hypotheses about the root cause of the interruption and selecting a new process model according to the best matching hypothesis.

In the event that the current data is insufficient to demarcate the models to be selected, the selection may be delayed until new measurement data arrives. The new measurement data may be included in the test, which may then allow to demarcate the models. If enough data is available, the best process model can be selected. Therefore, a measurement or any other way to obtain additional data may be initiated to enhance the selection of a suitable process model.

Such as In 17C is illustrated, the selected model 1706 for a further state estimation after the interruption 1700D be used. According to various embodiments, the state estimates may be necessary starting from the time point 1700T to which the interruption 1700D is detected, recalculated to obtain more accurate state information.

According to various embodiments, the modeling provided here of an interruption or any other disturbance in a complex and dynamic process, e.g. In a semiconductor manufacturing process, in a component of a control system and / or for monitoring discrete processes, as described above. According to various embodiments the mismatch between the process model (and according to the measurement model) and the real process data (and corresponding to the real measurement data) are monitored; wherein, when a mismatch is detected, two or more alternative models (also referred to as break hypotheses) may be tested, and based on the result of the testing, a process model that best matches the data may be selected. According to various embodiments, the process model may include the corresponding process equations and the measurement equations.

According to various embodiments, the data fusion circuit 108 be configured to determine a mismatch of the model. The acquired measurement data can z. B. be validated based on statistical error information.

A mismatch of the model can illustratively have several different reasons. According to various embodiments, the data fusion circuit may be configured to provide multiple reference models for mismatch of the model for various reasons, and to determine the reasons for mismatching the model by testing whether at least one of the reference models can properly model the acquired measurement data. If no reference models can be found that suitably model the acquired measurement data, e.g. B. the operation of the filter are interrupted and an error message can be sent. Alternatively, a reference model may be selected that fits best.

According to various embodiments, the data fusion circuit may be configured to select a first reference model from a plurality of reference models and recompute the state information based on the selected first reference model and determine a first model match parameter and select a second reference model from a plurality of reference models and the state information based on the selected second Recalculate the reference model and determine a second model match parameter and compare the first model match parameter with the second model match parameter and select the first or second model based on the comparison for further operation. The model match parameter may, according to various embodiments, include statistical error information, e.g. The normalized renewal of the relationship.

According to various embodiments, a method 1800 for controlling a processing arrangement, e.g. B. the process control circuit 100 , as described herein, as provided in U.S. Pat 18 is illustrated in a schematic process flow. According to various embodiments, the method 1800 Include: in 1810 Acquiring the process data and the measurement data of the processing device, wherein the process data includes a plurality of process event data according to a plurality of process events, and the measurement data includes a plurality of measurement event data according to a plurality of measurement events; in 1820 Providing state information representing a status of the processing device and accuracy information representing accuracy of the state information; in 1830 Updating the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and independently of updating the state information, restoring the prior state of the state information based on the acquired measurement event data, thereby increasing the accuracy of the state information, and 1840 Controlling the processing arrangement based on the state information.

• • um die Zustandsinformationen und die Genauigkeitsinformationen entsprechend bereitzustellen,
• • um die Zustandsinformationen entsprechend zu aktualisieren und entsprechend den früheren Zustand der Zusatzinformation wiederherzustellen und/oder
• • um die Bearbeitungsanordnung entsprechend zu steuern.

According to various embodiments, the method 1800 include one or more of the algorithms described herein and one or more of the methods described herein, e.g. B .:

• To provide the condition information and the accuracy information accordingly,
• • to update the state information accordingly and restore and / or restore the previous state of the additional information accordingly
• • to control the processing arrangement accordingly.

According to various embodiments, one or more aspects of the process control circuit 100 in software, eg Based on the MATLAB® software or any other suitable software. The algorithms described herein are provided in pseudocode to provide the main aspects to be implemented via any suitable programming environment.

According to various embodiments, a method may include using one or more of the algorithms described herein, e.g. To calculate one or more parameters or any other desired value, as described herein.

According to various embodiments, one or more of the above may be related to a process control circuit 100 or a data fusion circuit 108 also described as a method or as part of a method to implement a suitable system, e.g. B. a processing arrangement 104 to be controlled, understood.

Because the modeling of the processes may be correlated with the modeling of the measurements, within the description the terms "process model" or "model" may be used for the sake of brevity, such that they are e.g. B. corresponding to the process model and / or the measurement model mean. In the same way, the terms "process model parameters" or "model parameters" may be used for the sake of brevity, such that they are e.g. B. corresponding to the process model parameters and / or the measurement model parameters mean. In the same way, the term "process" can be used for the sake of brevity, so that it can be used e.g. B. means any aspect relating to the operation of the processing arrangement 104 must be taken into account, including z. As a machining tool and measuring tools. The term "data" can be used here, so that it z. B. any information means z. B. with regard to the processing arrangement 104 must be taken into account, including As the processing data, the measurement data, the tuning data and the like, such. Contextual / logistic data (a specific product, an intermediate mask, ...) and the data (process and measurement) of the previous process steps (including the preprocessing chamber / preprocessing tool, the pre-process (measurement) data, the contextual / logistic pre-process data, the pre-process processing time). Based on the operation of the algorithm, for. For example, additional data may be provided that controls the operation of the data merge circuit 108 can define, for. B. monitoring data, debugging data, tracking data, validating data, and the like. It will be appreciated that appropriate means for data processing, data storage and display of the data may be used as desired.

Hereinafter, various examples regarding the process control circuit and method for operating the process control circuit as described above and in a similar manner of a non-transitory computer-readable medium will be provided.

Several examples relate to the modeling and prediction of production processes (eg, batching processes) in a complex and dynamic manufacturing environment.

Example 1 is a process control circuit including: a data acquisition circuit configured to acquire process data and measurement data of a processing device, wherein the process data includes a plurality of process event data according to a plurality of process events and the measurement data includes a plurality of measurement event data corresponding to a plurality of measurement events; a data merging circuit configured to display state information, wherein the state information represents a status of the processing device, and accuracy information ( P providing an accuracy of the state information; wherein the data merging circuit is further configured to update the state information (eg, to perform a process update) based on the acquired process event data, thereby reducing the accuracy of the state information, and independently of updating the state information based on the acquired measurement event data, the previous state of the state information restore (eg, perform a metrology update), thereby increasing the accuracy of the state information, and a control circuit configured to configure the processing device based on the state information (FIG. X etc.) and / or the accuracy information .DELTA.X . P etc.).

Several examples relate to a process model for modeling and predicting production processes (eg, batching processes) in a complex and dynamic manufacturing environment.

In Example 2, the process control circuit of Example 1 optionally includes the data fusion circuit further configured to update the state information based on a process model that includes the process model parameters (eg, an accessible model parameter, A and or B ) and the process model estimation parameters (eg, an inaccessible model parameter, N and or Q ), to update.

In Example 3, the process control circuit of the example 1 or 2 optionally, the data fusion circuit is further configured to reconstruct the prior state of the state information based on a measurement model that includes the measurement model parameters (eg, an accessible measurement model parameter, C and or D ) and the measurement model estimation parameters (eg an inaccessible measurement model parameter, M and or R ), restore.

In example 4, the process control circuit according to any one of the preceding examples may optionally include the control circuit further configured to configure the processing arrangement based on both the state information (FIG. X etc.) as well as the accuracy information ( .DELTA.X . P etc.).

In example 5, the process control circuit according to any one of the preceding examples may optionally include the control circuit further configured to process control data to control the processing arrangement based on the state information (FIG. X etc.) and / or based on the accuracy information ( .DELTA.X . P etc.).

In Example 6, the process control circuit according to any one of the preceding examples may optionally include the control circuit further configured based on the state information (FIG. X ) and / or the accuracy information ( .DELTA.X . P etc.) to provide monitoring data. According to various embodiments, the monitoring data may be used to monitor at least one of the processing arrangement, the data acquisition circuit and / or the data merging circuit.

In Example 7, the process control circuit according to any one of the preceding examples may optionally include that the data merging circuit is configured to provide the accuracy information ( .DELTA.X ) based on a statistical estimation error (P, σ, σ ² , v, etc., as described herein).

In Example 8, the process control circuit according to any one of the preceding examples may optionally include the data merging circuit configured to merge the process data and the measurement data based on a Kalman filter to obtain the state information (FIG. X and .DELTA.X ) (or in other words appreciate).

In example 9, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to update the state information for all of the acquired process event data and for all of the acquired process measurement data in a chronological order defined by the process events and restore their previous state, wherein the measurement events are mapped to the process events.

In Example 10, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to update the state information for each of the plurality of process events in acquiring the corresponding process event data and for each of the plurality of measurement events in acquiring the corresponding measurement event data.

In Example 11, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to collect a plurality of process event data of the respective process events and to update the state information in consideration of the collected plurality of process event data upon acquiring the measurement event data of a subsequent trade event and the previous state Restore state information based on the acquired measurement event data. Illustratively, the process update may be executed at the latest when the measurement event data of the measurement event to be considered is available.

Several examples relate to the adaptation of recursive estimation and prediction algorithms to production scenarios with delayed and permuted measurements.

In Example 12, the process control circuit of any of the preceding examples may optionally include the data acquisition circuit further configured to correlate the measurement events with the corresponding process events (eg, mapping the measurement events to the corresponding process events). Therefore, based on at least one process event and at least one measurement event corresponding to the process event, a tuple may be provided. The tuple may be a product, for. A wafer, an electronic device, and the like.

In example 13, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to calculate current state information based on the process event data of a current process event and a sequence of previous process events and / or measurement events.

In example 14, according to one of the preceding examples, the process control circuit may optionally include the data fusion circuit configured to calculate current state information based on the process event data of a current process event and a sequence of previous process events (and eg previous measurement events) and capturing the process information Measurement event data corresponding to at least one previous process event of the sequence of previous process events, to recalculate the current condition information.

In example 15, the process control circuit of example 14 may optionally include the sequence of previous process events including a limited number of process events, starting from an initial process event defining the base state information and ending with the current process event defining the current state information.

In example 16, the process control circuit of example 15 may optionally include the initial process event being selected to be a first item in the sequence of the previous process events for which no corresponding measurement event data is available and for which the corresponding measurement event data is pending.

Several examples relate to a process model for a recursive estimation and prediction algorithm.

In example 17, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to process data based on a process model including one or more process model parameters for modeling known influences (A and / or B) and / or one or more Process estimation parameter for modeling unknown influences (N) contains, to model.

In example 18, the process control circuit of any one of the preceding examples may optionally include the data fusion circuit configured to configure the measurement data based on one or more measurement model parameters to model known influences (C and / or D) and / or one or more measurement estimation parameters to model unknown ones Influences (M) to model.

According to various embodiments, the one or more equations or the one or more algorithms for modeling the processes are independent of the one or more equations or the one or more algorithms for modeling the measurements.

In example 19, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to model the process data based on a process model (eg, a linear time-variant state space) that includes at least one of the following process model parameters: a state transition parameter (A), a process noise parameter (N) and / or a process drift parameter (B).

wobei x_k+1 ein Zustandsvektor ist, x_k ein vorhergehender Zustandsvektor ist, A_k+1 eine Zustandsübergangsmatrix ist, N_k+1 ein Prozessrauschvektor ist, B_k+1 ein Prozessdriftvektor ist und wobei k eine Zählervariable ist.In example 20, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to model the process data based on the following equation and / or a similar equation: $$x_{k+1}=A_{k+1}{x}_k+N_{k+1}+b_{k+1}.$$

where x _{k + 1 is} a state vector, x _{k is} a previous state vector, A _{k + 1 is} a state transition matrix, N _{k + 1 is} a process noise vector, B _{k + 1 is} a process drift vector, and where k is a counter variable.

In Example 21, the process control circuit of Example 20 may optionally include N _{k + 1} = Q _{k + 1} ξ _{k + 1} , where ξ _{k + 1 is} a random vector that reduces process noise (eg, with E [ξ] = 0, E [ξ ² ] = I), and is a Q _{k + 1} noise scaling matrix.

In example 22, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to model the measurement data based on a measurement model including at least one of the following measurement model parameters: an observation parameter (C), a measurement noise parameter (M), and / or a measurement offset parameter (D).

wobei y_k+1 ein skalarer Messwert oder ein Messvektor ist, C_k+1 ein Beobachtungsvektor oder eine Beobachtungsmatrix ist (der bzw. die den Beitrag der Elemente des Zustandsvektors x_k+1 zu y_k+1 beschreibt), M_k+1 ein Messrauschskalar oder ein Messrauschenvektor ist und D_k+1 ein Messversatzskalar oder ein Messversatzvektor ist und wobei k eine Zählervariable ist.In example 23, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to model the measurement data based on the following equation and / or a similar equation: $$y_{k+1}=C_{k+1}{x}_{k+1}+M_{k+1}+D_{k+1}.$$

where y _{k + 1 is} a scalar metric or measurement vector, C _{k + 1 is} an observation vector or observation matrix (which describes the contribution of elements of the state vector x _{k + 1} to y _{k + 1} ), M _{k + 1} is a measurement noise scalar or a measurement noise vector and D _{k + 1} is a measurement offset scalar or a measurement offset vector, and wherein k is a counter variable.

In Example 24, the process control circuit of Example 23 may optionally include M _{k + 1} = R _{k + 1} η _{k + 1} , where η _{k + 1 is} a scalar random value or a random vector representing the measurement noise (e.g. with E [η] = 0, E [η ² ] = I) and R _{k + 1 is} a noise scaling scalar or a noise scaling matrix.

In example 25, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to update the state information based on a first set of equations and to restore the previous state of the state information based on a second set of equations; of the first set of equations independently of the equations of the second set of equations.

$$P_{k+1}=A_{k+1}{P}_k{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T,$$

wobei P_k, P_k+1 eine Fehlerkovarianzmatrix (d.h., eine vorhergehende und aktualisierte Fehlerkovarianzmatrix) ist und x̂_k die Zustandsinformationen sind und x̂_k+1 die geschätzten Zustandsinformationen sind (d. h., x̂_k ist die vorhergehende und x̂_k+1 ist die aktualisierte Zustandsschätzung).Example 26 may optionally include the process control circuit of Example 25 having the first set of equations based on the following equations and / or similar equations: $${\widehat{x}}_{k+1}=A_{k+1}{\widehat{x}}_k+B_{k+1}.$$

$$P_{k+1}=A_{k+1}{P}_k{A}_{k+1}^T+Q_{k+1}{Q}_{k+1}^T.$$

where P _k , P _{k + 1 is} an error covariance matrix (ie, a previous and updated error covariance matrix) and x _{k are} the state information and x _{k + 1 are} the estimated state information (ie, x _k is the previous one and x _{k + 1} is the updated state estimate).

$$P_{k+1}=\left(I-K_{k+1}{C}_{k+1}\right)P_k$$

$$K_{k+1}=P_k{C}_{k+1}^T{\left(C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T\right)}^{-1},$$

wobei K_k+1 ein Rückkopplungsverstärkungsvektor ist und I eine Identitätsmatrix ist.In Example 27, the process control circuit of Example 25 or 26 may optionally include the second set of equations being based on the following equations and / or similar equations: $${\widehat{x}}_{k+1}={\widehat{x}}_k+K_{k+1}\left(y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}\right)$$

$$P_{k+1}=\left(I-K_{k+1}{C}_{k+1}\right)P_k$$

$$K_{k+1}=P_k{C}_{k+1}^T{\left(C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T\right)}^{-1}.$$

where K _{k + 1 is} a feedback gain vector and I is an identity matrix.

According to various embodiments, it is recognized that other equivalent formulations of the same equations can be obtained.

wobei P_k+1 eine Fehlerkovarianzmatrix ist und C ein Zustandsextraktorvektor ist.In Example 28, the process control circuit according to one of the preceding examples may optionally include the statistical error information of the estimated state information including a variance of the estimation error, σ ² _ERR , based on the following equation and / or a similar equation: $${\sigma }_{e\mathrm{<figure-callout\; id="2"\; label="R\; R"\; filenames="DE102017111926A1\_0254.png,DE102017111926A1\_0268.png"\; state="\{\{state\}\}">R\; </figure-callout>}R}^2=\overline{C}{P}_{k+1}{\overline{C}}^T.$$

where P _{k + 1 is} an error covariance matrix and C is a state extractor vector.

In example 29, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to provide predicted measurement event data (y _{k + 1} ) for a subsequent measurement event with corresponding (measurement) prediction error information.

In Example 30, the process control circuit of Example 29 may optionally include the (measurement) prediction error information of the predicted measurement event data being statistical error information.

In example 31, the process control circuit of example 29 or 30 may optionally include the predicted measurement event data being based on a combination of one or more elements of the estimated state vector.

In example 32, the process control circuit of any of examples 29 to 31 may optionally include the data fusion circuit further configured to validate the state information and / or a process model based on the accuracy information of the predicted measurement event data.

In example 33, the process control circuit of any of examples 29 to 32 may optionally include the data fusion circuit further configured to validate the state information and / or the process model based on the statistical error information of the predicted measurement event data.

In example 34, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to validate the acquired measurement data based on the accuracy information of the predicted measurement event data. The data fusion circuit may be configured to validate the acquired measurement data based on the statistical error information of the predicted measurement event data.

In example 35, the process control circuit of any one of the preceding examples may optionally include the data fusion circuit configured to validate the acquired measurement data by calculating a prediction error (also referred to as renewal).

In example 36, the process control circuit of example 35 may optionally include calculating the prediction error, v, based on the following equation and / or a similar equation: $$v=y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1},$$

In example 37, the process control circuit of any of the preceding examples may optionally include the data fusion circuit configured to validate the acquired measurement data by calculating a normalized renewal of the ratio and by comparing the calculated normalized renewal of the ratio to a reference value.

In example 38, the process control circuit of example 37 may optionally include calculating the normalized renewal of the ratio, v _o , based on the following equation and / or a similar equation: $$v_0=\frac{y_{k+1}-C_{k+1}{\widehat{x}}_k-D_{k+1}}{\sqrt{C_{k+1}{P}_k{C}_{k+1}^T+R_{k+1}{R}_{k+1}^T}},$$

In example 39, the process control circuit according to any one of the preceding examples may optionally include the data fusion circuit configured to use estimated measurement data a process model and a measurement model to check for mismatch between the measurement data and the estimated measurement data.

In example 40, the process control circuit of one of the preceding examples may optionally further include a communication circuit configured to provide (eg, generate and send) an alarm message in the event that a mismatch is detected according to example 39.

Several examples relate to coping with complex ambiguities.

In example 41, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to determine the accuracy information based on an estimation error, wherein the estimation error is decomposed into a finite part and an infinite part.

In example 42, the process control circuit of example 41 may optionally include determining the estimation error via an error covariance matrix (P) or an inverted error covariance matrix (P ^-1 , also referred to as an information matrix).

In Example 43, the process control circuit of Example 42 may optionally include the error covariance matrix or the inverted error covariance matrix decomposed into a finite part (P _R ) and an infinite part (P _I ).

mit κ → ∞.In example 44, the process control circuit of example 42 or 43 may optionally include decomposing the error covariance matrix (or the inverted error covariance matrix) according to the following equation and / or a similar equation: $$P_k=P_{R.k}+\kappa \cdot P_{I.k}.$$

with κ → ∞.

In example 45, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured, in a first case based on a first set of previous state recovery equations, and in a second case on a second set of recovery equations of the former state to restore the previous state of the state information, wherein the first case and the second case are distinguished from each other by a discrimination criterion.

In example 46, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured to calculate the prediction error for new measurements and to check whether the prediction error includes infinite parts.

In example 47, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured to calculate estimated measurement data and a corresponding prediction error and to check whether the prediction error includes infinite parts.

In example 48, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to restore the previous state of the state information for a measurement vector over a plurality of individual prior art scalar recoveries.

In example 49, the process control circuitry of example 48 may optionally include checking, for each of the individual scalar recoveries of the previous state, whether the corresponding measurement prediction error includes infinite parts.

In Example 50, the process control circuit of any of the preceding examples may optionally include the data fusion circuit configured to check based on a numeric reference, L _NUM , whether the measurement prediction error (or in other words, the new measurement value prediction error) is infinite Contains parts.

In diesem Fall entspricht F_I,k+1 dem unendlichen Teil des Vorhersagefehlers. In example 51, the process control circuit of example 50 may optionally include using a first set of previous state recovery equations to restore the previous state of state information if | F _{I, k + 1} | > L _NUM , and that a second set of previous state recovery equations is used to restore the previous state of state information if | F _{I, k + 1} | ≤ L _NUM , with: $$F_{I.k+1}:=C_{k+1}{P}_{l.k}{C}_{k+1}^T,$$

In this case, F _{I, k + 1} corresponds to the infinite part of the prediction error.

Illustratively, the observation parameter C in Example 51 selects a product and a measurement tool.

Several examples relate to a robust distinction between a finite and an infinite variance of the measured value prediction.

In example 52, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured to provide a discrimination value based on an observation parameter C and an error covariance matrix P and to compare the discrimination value with a reference value, L _NUM .

In example 53, the process control circuit of example 52 may optionally include the observation parameter C being factored.

mit C := C_k+1 und |C₀| = 1.In example 54, the process control circuit of example 53 may optionally include the observation parameter C an observation vector C which is factored based on the following equation and / or a similar equation: $$C={\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="DE102017111926A1\_0252.png,DE102017111926A1\_0257.png"\; state="\{\{state\}\}">C</figure-callout>}}_0\cdot \left|C\right|.$$

with C: = C _{k + 1} and | C ₀ | = 1.

Gemäß verschiedenen Ausführungsformen kann F_I0,k als ein Unterscheidungskriterium verwendet werden.The factorization according to Example 54 performs z. To the following $$F_{I0k}=C_0{P}_{I.k}{C}_0^{\text{T}},$$

According to various embodiments, F _{10, k may be used} as a distinguishing criterion.

Various examples relate to applying a square root factoring.

In example 55, the process control circuit according to any of the preceding examples may optionally include factoring the error covariance matrix or the inverted error covariance matrix (which is also referred to as the information matrix).

In example 56, according to one of the preceding examples, the process control circuit may optionally include the fact that the error covariance matrix or the inverted error covariance matrix is factored into triangular matrices. Optionally, the error covariance matrix or the inverted error covariance matrix may additionally be factored into one or more diagonal matrices (eg the matrix D).

In Example 57, the process control circuit of any of the preceding examples may optionally include the error covariance matrix (eg, P) (or the inverted error covariance matrix) being an n × n matrix (also referred to as an (n, n) matrix which has, for example, n rows and n columns) and that the triangular matrix (eg, S) is an n × n matrix, where n is a natural number. All factorization matrices of the factored error covariance matrix can be square and have the same size (n × n) as the error covariance matrix.

wobei D eine Diagonalmatrix ist. In example 58, the process control circuit of any one of the preceding examples may optionally include the error covariance matrix (or the inverted error covariance matrix) being factored based on the following equation and / or a similar equation: $$P=S^{T}S\text{or}P=U^{T}DU.$$

where D is a diagonal matrix.

In example 59, the process control circuit of any one of the preceding examples may optionally include the triangular matrix (eg, S or U) having a number of n rows and a rank r, and a number of n minus r rows that are exactly zero , contains. The error covariance matrix P has a number of n lines and is of rank r.

In example 60, the process control circuitry of example 59 may optionally include that for each of the n rows containing at least one non-zero element, the corresponding diagonal elements of the triangular matrix (eg, S or U) is not zero.

wobei D eine Diagonalmatrix ist.In example 61, the process control circuit of any one of the preceding examples may optionally include the fact that the error covariance matrix is factored based on the following equation and / or a similar equation: $$P=S{S}^T\text{or}P=UD{U}^T.$$

where D is a diagonal matrix.

According to various embodiments, the triangular matrix (eg, S or U) (eg, according to examples 58 and 61) may be an upper or lower right matrix or an upper or lower left matrix.

In example 62, the process control circuit of any of the preceding examples may optionally include the triangular matrix (eg, S or U) having a number of n columns and rank r and a number of n minus r columns that are exactly zero , contains. The error covariance matrix P has a number of n lines and is of rank r.

In example 63, the process control circuitry of example 62 may optionally include, for each of the n columns containing at least one non-zero element, the corresponding diagonal element of the triangular matrix (eg, S or U ) is not null.

In example 64, the process control circuit of any one of the preceding examples may optionally include the data fusion circuit further configured to have a number of g rows or columns that are exactly zero in the triangular matrix (e.g. S or U ) if the rank of the triangular matrix (e.g. S or U ) is reduced by the number g.

In example 65, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured to generate a number g of rows or columns that are exactly zero in the triangular matrix (eg, S or U) if the rank of the error covariance matrix P is reduced by the number g.

In example 66, the process control circuit of any of the preceding examples may optionally include the algorithm for restoring the previous state to include the rows or columns of the triangular matrix (e.g. S or U ) are processed sequentially (one after the other). Further, the rows / columns that are exactly zero can be processed by a procedure other than nonzero rows. Furthermore, existing zero rows / columns can be preserved.

In example 67, the process control circuit according to one of the preceding examples may optionally contain the finite part (FIG. P _R ) and the infinite part ( P _I ) of the error covariance matrix are factored. This can increase the numerical accuracy.

$$P_I=S_I^T{S}_I,P_I=S_I{S}_I^T,P_I=U_I^{T}D{U}_{\text{I }}\text{oder }{P}_I=U_{I}D{U}_I^T;$$

S_I S_R U_I U_R ein Cholesky-Faktor ist und wobei D eine Diagonalmatrix ist. In example 68, the process control circuit according to one of the preceding examples may optionally include the finite part (P _R ) and the infinite part ( P _I ) of the error covariance matrix are factored based on the following equation and / or a similar equation: $$P_R=S_R^T{S}_R.P_R=S_R{S}_R{}^T.P_R=U_R{}^{T}D{U}_R\text{or}{P}_R=U_{R}D{U}_R{}^T;$$

$$P_I=S_I^T{S}_I.P_I=S_I{S}_I^T.P_I=U_I^{T}D{U}_{\text{I}}\text{or}{P}_I=U_{I}D{U}_I^T;$$

S _I S _R U _I U _{R is} a Cholesky factor and where D is a diagonal matrix.

According to various embodiments, the Cholesky factor is a triangular matrix.

Various examples relate to a square root update.

In example 69, the process control circuit of any one of the preceding examples may optionally include the error covariance matrix (or, in a similar manner, the inverted error covariance matrix) decomposed.

In example 70, the process control circuit of any of the preceding examples may optionally include the error covariance matrix (or, in a similar manner, the inverted error covariance matrix) being factorized and decomposed.

In example 71, according to one of the preceding examples, the process control circuit may optionally include a finite part of the error covariance matrix being factored and decomposed, and an infinite part of the error covariance matrix being factored and decomposed.

oder $$P_R=U_{R}D{U}_R{}^T+P_{R\mathrm{,0}}.$$

In example 72, the process control circuit of any one of the preceding examples may optionally include the finite part of the error covariance matrix being factorized and decomposed based on the following equation and / or a similar equation: $$P_R=U_{R}D{U}_R{}^T+P_{R\mathrm{,\; 0}},$$

or $$P_R=U_{R}D{U}_R{}^T+P_{R\mathrm{,\; 0}},$$

oder $$P_I=U_{I}D{U}_I^T+P_{I\mathrm{,0}}$$

In example 73, the process control circuit of any one of the preceding examples may optionally include the infinite portion of the error covariance matrix being factorized and decomposed based on the following equation and / or a similar equation: $$P_I=U_{I}D{U}_I^T+P_{I\mathrm{,\; 0}}$$

or $$P_I=U_{I}D{U}_I^T+P_{I\mathrm{,\; 0}}$$

The matrices P _{R, 0} P _{I, 0} of Example 72 and 73 are diagonal matrices (of size n).

In example 74, the process control circuit of any of the preceding examples may optionally include decomposing the error covariance matrix into a main error covariance matrix and a process noise matrix to separately treat the accumulated process noise in the process noise matrix.

In example 75, the process control circuit of example 74 may optionally include the data fusion circuit further configured to introduce new process noise only to the process noise matrix to add (process update) and to transfer the (non-zero) elements of the process noise matrix to the main covariance matrix prior to any later restoration of the previous state in the case where the restoration of the previous state involves the state elements related to those matrix elements.

In example 76, the process control circuit of any one of the preceding examples may optionally include decomposing the inverted error covariance matrix (eg, the information matrix) into a main information matrix and a measurement noise matrix to separately treat the accumulated measurement noise in the measurement noise matrix.

In example 77, the process control circuit of example 76 may optionally include the data fusion circuit further configured to add new measurement noise only to the measurement noise matrix (restoration of the previous state) and the non-zero elements of the measurement noise matrix prior to any later process update to the main process. Transfer information matrix, if the update includes the state elements related to these matrix elements.

In example 78, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to update the state information based on a first set of update equations dealing with the state transition (A update) and a second set of update equations Add new process noise to update the state (Q update).

In Example 79, the process control circuit according to any one of the preceding examples may optionally include a state transition matrix (A) separated into unit blocks and non-unit blocks to update the state information.

In example 80, the process control circuit of example 78 or 79 may optionally include the state transition matrix (FIG. A ) is formulated as a block diagonal matrix having a plurality of unit blocks and non-unit blocks, or transformed into a block diagonal matrix having a plurality of unit blocks and non-unit blocks.

In example 81, the process control circuit of example 79 or 80 may optionally include the overall process update being formulated as a sequence of sub-process updates, with each sub-process update taking into account one of the multiple non-entity blocks.

Several examples relate to the validation of Kalman filter models tuned to the actual data.

In example 82, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to include the process model (eg, the process model parameters) and / or the measurement model (eg, the measurement model parameters) based on a set of training data to validate and / or vote.

In example 83, the process control circuitry of example 82 may optionally include the set of training data containing the available process data (eg, the context data and / or the logistics data of the processing arrangement and operation) and / or the available measurement data (e.g. B. are mapped to the process data) contains.

In example 84, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to validate a tuned process model and / or a tuned measurement model.

In example 85, the process control circuit of example 84 may optionally include the data fusion circuit configured to further configure the (eg, tuned) process model and / or the (eg, tuned) measurement model based on a normalized prediction error signal to predict reference values. Validate measurement event data for multiple measurement events.

In example 86, the process control circuit of any one of the preceding examples may optionally include the data fusion circuit further configured to depict the process model and / or the process model Validating a measurement model based on an average value that takes into account multiple symmetrized normalized prediction errors.

In example 87, the process control circuit of example 86 may optionally include providing the symmetrized normalized prediction errors by applying a symmetric function to the normalized prediction errors.

In example 88, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to configure the process model and / or measurement model (eg, all parameters) based on a probability signal and / or a probability gradient signal for one or more model parameters (eg for a specific model parameter). The signal may be a sequence of numbers, each number corresponding to a reading.

Various examples relate to tracking, monitoring, and debugging a recursive estimation and prediction algorithm.

In example 89, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to store a plurality of backup state information (to thereby generate, for example, additional historical data) and a plurality of backup model parameters to monitor at least one state information is to reconstruct.

In example 90, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to reconstruct at least one state information to be monitored during operation of the filter.

In example 91, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured to include the at least one state information to be monitored by recalculating the at least one state information to be monitored, starting from one of the stored ones to reconstruct multiple backup state information and considering or incorporating one or more of the stored multiple model parameters.

In example 92, the process control circuit of one of the preceding examples may optionally further include a data processing circuit configured to reconstruct (eg, a user-defined) combination of one or more state information elements of the at least one state information to be monitored.

In example 93, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to decompose state information or a combination of state information elements of the state information into a weighted sum taking into account a plurality of previous measurement event data.

In example 94, the process control circuit of one of the preceding examples may optionally include the data fusion circuit further configured to provide one or more tracking and / or debug metrics considering at least one user-defined (eg, linear) combination of the state information elements of the state information.

In example 95, the process control circuit of any of the preceding examples may optionally include the data fusion circuit further configured to calculate an exact probability gradient of the process model (eg, for the process equations and / or the measurement equations).

In example 96, the process control circuit of example 95 may optionally include using a generalized inverse (pseudo inverse) of a product to calculate an exact probability gradient of the process model, the product including a matrix of a process or measurement model parameter and the transpose of the matrix.

Gemäß verschiedenen Ausführungsformen ist die Matrix N im Beispiel 97 faktorisiert. In example 97, the process control circuit of example 95 or 96 may optionally include the data fusion circuit further configured to calculate an exact probability gradient of the process model, using the following equation or a similar equation: $$N_{k-1}=\left(I-C_k^T{K}_k^T\right)A_{k+1}^T{N}_k{A}_{k+1}\left(I-C_k{K}_k\right)+C_k^T{F}_k^{-1}{C}_k,$$

According to various embodiments, the matrix N is factored in Example 97.

In example 98, the process control circuit according to one of the preceding examples may optionally include the data fusion circuit 108 further configured to provide an exact probability gradient of the process model and / or the measurement model using a generalized inverse of a product, wherein the product includes a matrix of a process or measurement model parameter and its transpose, and / or using a forward and a backward iteration at least one iteration matrix in the forward iteration is factored and / or wherein at least one iteration matrix is factored in the backward iteration.

In example 99, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to provide a unified process model parameter for one or more of the process model parameters, respectively. The unified process model parameter represents multiple process model parameters of the corresponding process events for which no measurement events are considered.

Various examples relate to interrupt modeling for a process model and / or measurement model for a recursive estimation and prediction algorithm.

In example 100, the process control circuit of any of the preceding examples may optionally include the data fusion circuit configured to validate the acquired measurement data based on statistical error information to determine a model mismatch.

In example 101, the process control circuit according to any one of the preceding examples may optionally include the data fusion circuit configured, e.g. In the case where a mismatch is determined, as described in Example 100, select a reference model from a plurality of reference models and recompute the state information based on the selected reference model.

In example 102, the process control circuit of one of the preceding examples may optionally include the data fusion circuit configured to select a first reference model from a plurality of reference models and recalculate the state information based on the selected first reference model and determine a first model match parameter and a second reference model select multiple reference models and recalculate the state information based on the selected second reference model and determine a second model match parameter and compare the first model match parameter with the second model match parameter and select the first or the second model for further operation based on the comparison.

In example 103, the process control circuit according to any one of the preceding examples may optionally include the data fusion circuit further configured to use a model used to update and / or restore the previous state (eg, the process model and / or the measurement model). to validate and / or vote.

Example 104 is a method of controlling a processing arrangement, the method including: acquiring process data and measurement data of the processing device, wherein the process data includes a plurality of process event data according to a plurality of process events and the measurement data includes a plurality of measurement event data according to a plurality of measurement events; Providing state information representing a status of the processing device and accuracy information representing accuracy of the state information; Updating the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and independently of updating the state information, restoring the previous state of the state information based on the acquired measurement event data, thereby increasing the accuracy of the state information, and controlling the processing arrangement based on the Status information and / or the accuracy information.

Example 105 is a non-transitory computer-readable medium having program instructions that, when executed, cause a device to perform control of a processing arrangement including: acquiring process data and measurement data of the processing device, wherein the process data includes a plurality of process event data according to a plurality of process events; the measurement data includes a plurality of measurement event data in accordance with a plurality of measurement events; Providing state information representing a status of the processing device and accuracy information representing accuracy of the state information; Updating the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and independently of updating the state information, restoring the previous state of the state information based on the acquired measurement event data, thereby increasing the accuracy of the state information, and controlling the processing arrangement based on the Status information and / or the accuracy information.

Example 106 is a process control circuit including: a data acquisition circuit configured to capture the process data and the measurement data of a processing device, the process data including a plurality of process event data according to a plurality of process events and the measurement data including a plurality of measurement event data according to a plurality of measurement events; a data merging circuit configured to provide estimated state information, wherein the estimated state information represents an estimated state of the processing device, and to provide error information representing an estimation error of the estimated state information; wherein the data merging circuit is further configured to provide the estimated state information by updating and restoring the prior state of previous state information based on the acquired process event data and measurement event data, wherein the estimation error is increased by updating (eg, taking into account the process data) and wherein the estimation error is reduced by restoring the previous state (eg, taking into account the measurement event data), and a control circuit configured to configure the processing arrangement based on the estimated state information ( X ) and / or the estimation error.

Example 107 is a process control circuit including: a data acquisition circuit configured to capture the process data and the measurement data of a processing device, the process data including a plurality of process event data according to a plurality of process events and the measurement data including a plurality of measurement event data according to a plurality of measurement events; A recursive process control circuit configured to recursively calculate current state information based on the process event data of a current process event and a sequence of previous process (and eg measurement) events and the current state information in acquiring measurement event data that is at least one previous process event corresponding to the sequence of previous process events, (and / or recalculating process event data), and a control circuit configured to configure the processing arrangement based on the current state information ( X ) and / or based on accuracy ( .DELTA.X ) of the current status information ( X ) to control.

In Example 108, the process control circuit of any one of the preceding examples is used in batching to fabricate a plurality of products, each of the plurality of products being fabricated via multiple processes. Lottery production can lead to a huge amount of data that needs to be handled, with the vast amount of data being stored by the data fusion circuitry described herein 108 is treated automatically.

While the invention has been particularly shown and described with respect to specific embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as set forth in the appended claims Claims are defined to deviate. The scope of the invention is, therefore, indicated by the appended claims, it is therefore intended that all changes coming within the meaning and range of equivalency of the claims be included.

Claims (29)

A process control circuit (100) comprising: a data acquisition circuit (102) configured to capture the process data and the measurement data of a processing device (104), the process data including a plurality of process event data according to a plurality of process events and the measurement data including a plurality of measurement event data according to a plurality of measurement events; a data merge circuit (108) configured to provide state information, wherein the state information represents a status of the processing device (104), and provide accuracy information representing an accuracy of the state information; wherein the data fusion circuit (108) is further configured to update the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and to restore the prior state of state information independently of updating the state information based on the acquired measurement event data Status information, and a control circuit (110) configured to control the processing device (104) based on the state information and / or based on the accuracy information. Process control circuit after Claim 1 wherein the data merging circuit (108) is further configured to update the state information based on a process model taking into account the process model parameters and the process model estimation parameters; and / or wherein the data fusion circuit is further configured to recover the previous state of the state information based on a measurement model that takes into account the measurement model parameters and the measurement model estimation parameters. Process control circuit after Claim 1 or 2 wherein the control circuit (110) is further configured to provide process control data based on the state information and / or based on the accuracy information to control the processing device (104); and / or wherein the control circuit is further configured to provide monitoring data based on the state information and / or based on the accuracy information to monitor at least one of the processing device (104), the data acquisition circuit (102), or the data merging circuit (108). Process control circuit according to one of Claims 1 to 3 wherein the data merging circuit (108) is configured to provide the accuracy information based on a statistical estimation error. Process control circuit according to one of Claims 1 to 4 wherein the data merging circuit (108) is configured to merge the process data and the measurement data based on a Kalman filter to provide the state information and / or the accuracy information. Process control circuit according to one of Claims 1 to 5 wherein the data acquisition circuit (102) is further configured to correlate the measurement events with the corresponding process events. Process control circuit according to one of Claims 1 to 6 wherein the data fusion circuit (108) is further configured to calculate current state information based on the process event data of a current process event and a sequence of previous process events, and to recalculate the current state information upon acquisition of measurement event data corresponding to at least one previous process event of the sequence of previous process events , Process control circuit according to one of Claims 1 to 7 . wherein the data merging circuit (108) is further configured to model the process data based on a process model that includes one or more process model parameters for modeling known influences and / or one or more process estimation parameters for modeling unknown influences; and wherein the data fusion circuit (108) is further configured to model the measurement data based on one or more measurement model parameters for modeling known influences and / or one or more measurement estimation parameters for modeling unknown influences. Process control circuit according to one of Claims 1 to 8th wherein the data merging circuit (108) is further configured to update the state information based on a first set of equations and restore the previous state of the state information based on a second set of equations, the equations of the first set of equations being independent of the equations of second set of equations. Process control circuit according to one of Claims 1 to 9 wherein the data fusion circuit (108) is configured to provide estimated measurement data using a process model and a measurement model to check for a mismatch between the measurement data and the estimated measurement data. Process control circuit after Claim 10 further comprising: a communication circuit (312) configured to provide an alarm message or an alarm signal in the event a mismatch is detected. Process control circuit according to one of Claims 1 to 11 wherein the data merging circuit (108) is further configured to determine the accuracy information based on an estimation error, wherein the estimation error is decomposed into a finite part and an infinite part. Process control circuit after Claim 12 wherein the finite part of the estimation error is determined via a factorized error covariance matrix or a factored inverted error covariance matrix and / or wherein the infinite part of the estimation error is determined via another factorized error covariance matrix or another factorized inverted error covariance matrix. Process control circuit according to one of Claims 1 to 13 wherein the data fusion circuit (108) is further configured to restore the previous state of the state information based on a first set of previous state recovery equations in a first case and on a second set of earlier state restore equations in a second case, wherein the first case and the second case are distinguished from one another by a distinguishing criterion. Process control circuit according to one of Claims 1 to 14 wherein the data fusion circuit (108) is further configured to restore the previous state of the state information based on a first set of previous state recovery equations in a first case and on a second set of earlier state restore equations in a second case, wherein the first case and the second case are differentiated from each other by a discrimination value, wherein the data merging circuit (108) is further configured to provide the discrimination value based on an observation vector and an error covariance matrix and compare the discrimination value with a reference value, wherein the observation vector is factorized. Process control circuit according to one of Claims 1 to 15 wherein the data merging circuit (108) is further configured to determine the accuracy information based on an estimation error, and wherein the estimation error is determined via a factorized error covariance matrix or a factorized inverted error covariance matrix. Process control circuit after Claim 16 wherein the error covariance matrix or the inverted error covariance matrix is factored into at least two triangular matrices, wherein at least one triangular matrix of the at least two triangular matrices has a number n of columns of rank r and comprises a number of n minus r columns which are exactly zero; wherein for each of the n columns comprising at least one element other than zero, the corresponding diagonal element of the at least one triangular matrix is not zero. Process control circuit according to one of Claims 1 to 17 wherein the data merging circuit (108) is further configured to determine the accuracy information based on an estimation error, wherein the estimation error is determined via an error covariance matrix or an inverted error covariance matrix; and wherein the error covariance matrix is decomposed into a main covariance matrix and into a process noise matrix to separately treat the accumulated process noise in the process noise matrix, or wherein the inverted error covariance matrix is decomposed into an inverted main covariance matrix and a measurement noise matrix to separate the accumulated measurement noise separately to treat the measurement noise matrix. Process control circuit after Claim 18 wherein the data merging circuit is further configured to add new process noise only to the process noise matrix and the elements of the process noise matrix prior to any later restoration of the previous state in the case where the later restoration of the previous state involves the state elements related to those matrix elements Covariance matrix to transmit; or wherein the data fusion circuit is further configured to add new measurement noise only to the measurement noise matrix and to transmit the elements of the measurement noise matrix to the main inverted covariance matrix prior to any subsequent update in the case where the later update includes the state elements related to those matrix elements. Process control circuit according to one of Claims 1 to 19 wherein the data merging circuit (108) is further configured to update the state information based on a process model and to validate and / or tune the process model and / or the measurement model based on a set of training data. Process control circuit according to one of Claims 1 to 20 wherein the data merging circuit (108) is further configured to store a plurality of save state information and a plurality of save model parameters to reconstruct at least one state information to be monitored. Process control circuit according to one of Claims 1 to 21 wherein the data merging circuit (108) is further configured to provide an exact probability gradient of the process model and / or the measurement model using a generalized inverse of a product, the product comprising a matrix of a process or measurement model parameter and its transpose; and / or using a forward and backward iteration, wherein at least one iteration matrix is factored in the forward iteration and / or wherein at least one iteration matrix is factored in the backward iteration. Process control circuit according to one of Claims 1 to 22 wherein the data merging circuit (108) is further configured to update the state information based on a unified process model parameter, wherein the unified process model parameter represents a plurality of process model parameters of the corresponding process events for which no measurement events are considered. Process control circuit according to one of Claims 1 to 23 wherein the data merging circuit (108) is configured to validate the acquired measurement data based on the accuracy information to determine a model mismatch, and in the case where a mismatch of the model is determined to select a reference model from a plurality of reference models; Recalculate state information based on the selected reference model. Process control circuit according to one of Claims 1 to 24 wherein the data fusion circuit (108) is configured to update the state information based on a state transition matrix formulated as a block diagonal matrix having a plurality of unit blocks and a plurality of non-unit blocks, or transformed into a block diagonal matrix having a plurality of unit blocks and a plurality of nonunit blocks, and wherein a total process update as one Sequence of sub-process updates, each of the sub-process updates taking into account one of the multiple non-unit blocks. Process control circuit according to one of Claims 1 to 25 wherein the data merging circuit (108) is configured to validate the process model and / or the measurement model based on at least one of the following: a normalized prediction error signal; an average value taking into account a plurality of symmetrized normalized prediction errors; a probability signal; and or a probability gradient signal. A method of controlling a processing arrangement, the method comprising: Acquiring process data and measurement data of the processing device, wherein the process data includes a plurality of process event data according to a plurality of process events, and the measurement data includes a plurality of measurement event data according to a plurality of measurement events; Providing state information representing a status of the processing device and accuracy information representing accuracy of the state information; Updating the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and regardless of updating the state information Restoring the previous state of the state information based on the acquired measurement event data, thereby increasing the accuracy of the state information, and Controlling the processing arrangement based on the state information and / or the accuracy information. A non-transitory computer-readable medium having program instructions that, when executed, cause a device to perform control of a processing arrangement comprising: Acquiring process data and measurement data of the processing device, wherein the process data includes a plurality of process event data according to a plurality of process events, and the measurement data includes a plurality of measurement event data according to a plurality of measurement events; Providing state information representing a status of the processing device and accuracy information representing accuracy of the state information; Updating the state information based on the acquired process event data, thereby reducing the accuracy of the state information, and regardless of updating the state information Restoring the previous state of the state information based on the acquired measurement event data, thereby increasing the accuracy of the state information, and Controlling the processing arrangement based on the state information and / or the accuracy information. Using a process control circuit according to one of Claims 1 to 26 in a batch process for making multiple products, each of the multiple products being manufactured through multiple processes.

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